GF(2m)椭圆曲线上的Co_Z点加算法

GF ( <math id="M1"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118297&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118295&type= 6.77333355 6.01133347 )椭圆曲线上的Co_ Z 点加算法 Co_ Z Addition on Elliptic Curves over Finite Fields GF ( <math id="M2"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118306&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118303&type= 4.48733330 3.97933340 ) 1 first-author 刘双根，男，副教授，博士，主要研究领域为密码学与信息安全。E⁃mail:liusgxupt@163.com 刘双根，男，副教授，博士，主要研究领域为密码学与信息安全。E⁃mail:liusgxupt@163.com liusgxupt@163.com 1 1 1 西安邮电大学 通信与信息工程学院，西安 陕西 710121 西安邮电大学 通信与信息工程学院，西安 陕西 710121 School of Communication and Information Engineering， Xi’an University of Posts and Telecommunications， Xi’an 710121， Shaanxi， China School of Communication and Information Engineering， Xi’an University of Posts and Telecommunications， Xi’an 710121， Shaanxi， China 基于统一 Z 坐标，提出了有限域 GF ( <math id="M3"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118314&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118310&type= 3.04799986 2.62466669 )上两种射影坐标下的Co_Z点加运算公式。通过对椭圆曲线上有理点的 Z 坐标统一化处理，使得其运算量分别为10M+3S和8M+3S(M,S分别表示有限域上的乘法和平方)，相比已有的计算公式，运算量分别减少了2S和2M。另外，提出了一种Jacobian坐标下的3 P 运算公式，运算量减少了1M+4S。将新提出的点加、3 P 运算公式与对称三进制标量乘相结合，改进了标量乘算法的运算效率，使得标量乘算法的效率提高了13%。 Based on same Z ⁃coordinate, Co_ Z point addition formulae for two kinds of projective coordinates over finite field GF ( <math id="M4"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118323&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118319&type= 3.04799986 2.62466669 ) are proposed. Using the unified processing of the Z coordinate of the rational points on the elliptic curve, one point addition operation cost is 10M+3S and the other is 8M+3S (M stands for multiplication, and S stands for square on the finite field). Compared with the existing algorithms, the costing of operation is reduced by 2S and 2M, respectively. In addition, a formula for 3 P operation in Jacobian coordinate is proposed, which reduces the computational complexity by 1M+4S. Combining the new formulae with symmetric ternary scalar multiplication, the operational efficiency of the scalar multiplication is improved 13%. 统一 Z 运算 射影坐标 对称三进制 Co_ Z operation projective coordinate symmetric ternary 0　 引 言 1 自20世纪80年代中期椭圆曲线密码体制(elliptic curve cryptography, ECC) [ 1 1 , 2 2 ] 被提出之后，由于其密钥短，运算量相对较少，安全性能高等优势，已被广泛应用在资源受限的硬件设备中 [ 3 3 ] 。ECC的安全性依赖于离散对数问题的难解性，如果攻击者能很容易从用户的公钥中计算出私钥，那么这个密码体制就是不安全的。在ECC中，标量乘算法 kP （ k 为用户的私钥， P 为用户选取的椭圆曲线上的一有理点）是最耗时，也是最关键的算法。当攻击者截获或根据消耗轨迹推算出标量 k ，就能破解椭圆曲线加密，因此，标量乘的安全决定着整个椭圆曲线密码体制的安全性。而标量乘的运算易受到侧信道攻击(side channel attack, SCA) [ 4 4 ] 。 标量乘算法的运行速度决定着整个ECC的实现效率，因此通过提高标量乘的运行速度，可以提高ECC的效率。标量乘算法主要包括两个层次：上层标量 k 的展开；底层有限域内点加、倍点的域乘、域平方、求逆等基础运算 [ 5 5 ] ，本文重点研究底层。Co_ Z 点加运算在2007年由Meloni [ 6 6 ] 首次提出，该运算通过统一椭圆曲线上两射影点的 Z 坐标，使点加运算量降低，得出了新的点加算法，并将其应用在Euclidean加法链和Zeckendorf展开式中，通过减少点加运算中的域乘和域平方计算次数提高标量乘效率。近几年，Goundar、Wei等 [ 7 7 , 8 8 ] 将Co_ Z 算法应用到Montgomery阶梯算法、Joy倍点点加算法和NAF(non⁃adjacent form)算法，优化了Weierstrass椭圆曲线上的标量乘运算公式。2017年，于伟等 [ 9 9 ] 在Montgomery算法的基础上，首次利用统一 Z 坐标技巧，改进了有限域 GF ( <math id="M5"><msup><mrow><mn mathvariant="normal">3</mn></mrow><mrow><mi>m</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118332&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118327&type= 3.38666677 3.04799986 )上椭圆曲线的点加和倍点公式。本文，我们提出特征2有限域椭圆曲线上的Co_ Z 点加运算，推导出不同射影坐标系下的点加及三倍点（3 P ）运算公式。 1　 背景知识 1 1.1　有限域 GF ( <math id="M6"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118343&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118338&type= 3.38666677 3.04799986 )上的椭圆曲线 2 定义在域 K 上的一个Weierstrass方程 E ： <math id="M7"><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>x</mi><mi>y</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi>y</mi><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msub><mi>x</mi><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118349&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118347&type= 63.50000000 4.91066647 通过变量的相容性，可以简化方程。根据域 K 的特征不同，将椭圆曲线分为3种情况：特征不等于2或3、特征等于2和特征等于3。 当域 K 的特征不等于2或不等于3时，通过相容性变换，曲线 E 可简化为： <math id="M8"><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mi>a</mi><mi>x</mi><mo>+</mo><mi>b</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118357&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118353&type= 28.44799995 3.97933340 其中 <math id="M9"><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118365&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118360&type= 13.37733269 3.21733332 。曲线的判别式是 <math id="M10"><mi mathvariant="normal">Δ</mi><mo>=</mo><mo>-</mo><mn mathvariant="normal">16</mn><mo stretchy="false">(</mo><mn mathvariant="normal">4</mn><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mn mathvariant="normal">27</mn><msup><mrow><mi>b</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118373&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118368&type= 34.29000092 3.81000018 。 若域 K 的特征是3，则其非超奇异椭圆曲线 [ 10 10 ] 简化为： <math id="M11"><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118380&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118377&type= 15.49400043 3.97933340 <math id="M12"><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118383&type= 11.76866722 3.13266659 ，其中 <math id="M13"><mi>a</mi><mo>,</mo><mi>b</mi><mo>∈</mo><mi>K</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118391&type= 13.37733269 3.21733332 ， <math id="M14"><mi mathvariant="normal">Δ</mi><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118407&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118403&type= 6.77333355 2.79399991 <math id="M15"><mo>-</mo><msup><mrow><mi>a</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118413&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118410&type= 8.88999939 3.04799986 。 本文研究的重点是特征为2的非超奇异椭圆曲线，其Weierstrass方程可简化为： <math id="M16"><msup><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>x</mi><mi>y</mi><mo>=</mo><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mi>a</mi><msup><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118422&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118418&type= 36.32199860 3.97933340 ， <math id="M17"><mi mathvariant="normal">Δ</mi><mo>=</mo><mi>b</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118431&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118427&type= 9.14400005 2.79399991 。 根据“弦和切线”法则 [ 11 11 ] ，有限域 GF ( <math id="M18"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118437&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118434&type= 3.38666677 3.04799986 )上的两点相加得到椭圆曲线上的第三点。椭圆曲线上的点集合及这种加法运算构成了一个加法交换群，∞为加法交换群的无穷远点。因此，有限域 GF ( <math id="M19"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118445&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118442&type= 3.38666677 3.04799986 )上的非超奇异椭圆曲线的群运算法则为： 1） 单位元 3 对于所有的 <math id="M20"><mi>P</mi><mo>∈</mo><mi>E</mi><mo stretchy="false">(</mo><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118452&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118450&type= 25.14599991 3.81000018 ， <math id="M21"><mi>P</mi><mo>+</mo><mi mathvariant="normal">∞</mi><mo>=</mo><mi mathvariant="normal">∞</mi><mo>+</mo><mi>P</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118459&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118456&type= 26.66999817 2.96333337 2） 负元素 3 如果 <math id="M22"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>E</mi><mo stretchy="false">(</mo><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118467&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118463&type= 39.20066452 4.06400013 ，则 <math id="M23"><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118475&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118471&type= 19.38866615 3.81000018 <math id="M24"><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo><mo>=</mo><mi mathvariant="normal">∞</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118483&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118479&type= 18.45733452 3.81000018 。记点 <math id="M25"><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>x</mi><mo>+</mo><mi>y</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118490&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118486&type= 16.42533302 3.81000018 为- P ，并称其为 P 的负。注意，- P 也是 <math id="M26"><mi>E</mi><mo stretchy="false">(</mo><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118498&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118494&type= 18.11866570 3.81000018 上的一点。 3） 点加 3 令 <math id="M27"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>E</mi><mo stretchy="false">(</mo><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118507&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118504&type= 40.80933380 4.99533367 , Q =( x 2 , y 2 )∈ E ( GF (2 m ))， <math id="M28"><mi>P</mi><mo>≠</mo><mi>Q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118516&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118512&type= 10.83733368 3.30199981 ,则 <math id="M29"><mi>P</mi><mo>+</mo><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118525&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118520&type= 25.90799904 4.74133348 。其中 <math display="block" id="M30"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfrac><mrow><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>a</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118532&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118529&type= 67.22533417 20.06599998 4） 倍点 3 令 <math id="M31"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>E</mi><mo stretchy="false">(</mo><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118542&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118537&type= 40.80933380 4.99533367 ， <math id="M32"><mi>P</mi><mo>≠</mo><mo>-</mo><mi>P</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118551&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118546&type= 14.13933372 3.04799986 ,则 <math id="M33"><mn mathvariant="normal">2</mn><mi>P</mi><mo>=</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118557&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118555&type= 8.63599968 2.79399991 <math id="M34"><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118562&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118559&type= 11.17599964 4.74133348 。其中 <math display="block" id="M35"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mo>+</mo><mfrac><mrow><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mfrac><mrow><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo>+</mo><mi>a</mi><mo>=</mo><msup><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfrac><mrow><mi>b</mi></mrow><mrow><msup><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac></mtd></mtr><mtr><mtd><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mfrac><mrow><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo stretchy="false">)</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118571&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118566&type= 71.62799835 20.06599998 由此可得出，仿射坐标系下，点加运算量为1次求逆(I)，1次平方(S)和2次乘(M)；倍点运算量为1I+1S+2M。由于求逆的计算代价较高，因此通过转换坐标系，避免运算公式中出现求逆，可降低点加和倍点的运算代价 [ 12 12 ] 。 1.2　射影坐标下的点加和倍点运算公式 2 本文中，我们考虑两种射影坐标下的点加和倍点运算。 1.2.1　López & Dahab射影坐标 3 射影点 <math id="M36"><mo stretchy="false">(</mo><mi>X</mi><mo>:</mo><mi>Y</mi><mo>:</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118577&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118574&type= 13.54666710 3.55599999 , <math id="M37"><mi>Z</mi><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118588&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118582&type= 9.82133293 3.04799986 与仿射点 <math id="M38"><mo stretchy="false">(</mo><mfrac><mrow><mi>X</mi></mrow><mrow><mi>Z</mi></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>Y</mi></mrow><mrow><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118599&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118592&type= 14.13933372 7.78933334 对应， GF ( <math id="M39"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118610&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118606&type= 3.38666677 3.04799986 )上的Weierstrass方程的投影形式定义为 <math display="block" id="M40"><msup><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>X</mi><mi>Y</mi><mi>Z</mi><mo>=</mo><msup><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mi>Z</mi><mo>+</mo><mi>a</mi><msup><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118621&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118615&type= 53.67866516 3.13266659 令 <math id="M41"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118630&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118624&type= 23.53733444 4.48733330 , <math id="M42"><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118640&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118633&type= 24.04533386 4.48733330 , <math id="M43"><mi>P</mi><mo>+</mo><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118648&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118644&type= 31.24200058 4.48733330 。若 <math id="M44"><mi>P</mi><mo>≠</mo><mi>Q</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118658&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118653&type= 10.83733368 3.30199981 ,则点加运算公式为 <math display="block" id="M45"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>G</mi><mi>I</mi><mo stretchy="false">(</mo><mi>H</mi><mo>+</mo><msup><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>a</mi><mi>G</mi><mi>I</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mi>G</mi><mi>H</mi><mi>I</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><mi>A</mi><mi>G</mi><msup><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><mi>E</mi><msup><mrow><mi>I</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mi>G</mi><mi>I</mi><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118666&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118662&type= 63.92333221 15.74800014 其中： <math id="M46"><mi>A</mi><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118674&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118671&type= 15.32466698 3.72533321 ， <math id="M47"><mi>B</mi><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118682&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118677&type= 14.98599911 3.72533321 ， <math id="M48"><mi>C</mi><mo>=</mo><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118689&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118685&type= 12.44600010 3.97933340 ， <math id="M49"><mi>D</mi><mo>=</mo><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118696&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118693&type= 12.69999981 3.97933340 ， <math id="M50"><mi>E</mi><mo>=</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118702&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118699&type= 14.73199940 3.72533321 ， <math id="M51"><mi>F</mi><mo>=</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi>C</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118709&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118705&type= 14.22399998 3.72533321 ， <math id="M52"><mi>G</mi><mo>=</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118715&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118712&type= 15.24000072 3.72533321 ， <math id="M53"><mi>H</mi><mo>=</mo><mi>E</mi><mo>+</mo><mi>F</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118721&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118718&type= 17.77999878 2.96333337 ， <math id="M54"><mi>I</mi><mo>=</mo><mi>A</mi><mo>+</mo><mi>B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118730&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118725&type= 16.25600052 2.96333337 。其倍点运算公式为 <math display="block" id="M55"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr><mtr><mtd><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mi>C</mi><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mi>C</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118737&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118733&type= 57.06533432 14.39333344 其中： <math id="M56"><mi>A</mi><mo>=</mo><msup><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>,</mo><mi>B</mi><mo>=</mo><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>,</mo><mi>C</mi><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118744&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118739&type= 44.28066635 4.40266657 。 根据运算公式，可以看出一次点加运算量为13M+5S，一次倍点运算量为4M+5S。当假设 <math id="M57"><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118752&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118748&type= 10.41399956 3.72533321 时，点加运算量降至10M+5S。López等 [ 13 13 ] 改进了倍点运算，将运算量降低到3M+5S。 1.2.2　Jacobian射影坐标 3 射影点 <math id="M58"><mo stretchy="false">(</mo><mi>X</mi><mo>:</mo><mi>Y</mi><mo>:</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118760&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118757&type= 13.54666710 3.55599999 , <math id="M59"><mi>Z</mi><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118765&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118762&type= 9.82133293 3.04799986 与仿射点 <math id="M60"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><mfrac><mrow><mi>X</mi></mrow><mrow><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>Y</mi></mrow><mrow><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118773&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118769&type= 22.43666649 7.78933334 对应， GF ( <math id="M61"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118780&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118777&type= 3.38666677 3.04799986 )上的Weierstrass方程的投影形式定义为 <math display="block" id="M62"><msup><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>X</mi><mi>Y</mi><mi>Z</mi><mo>=</mo><msup><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mi>a</mi><msup><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">6</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118787&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118784&type= 51.05400085 3.13266659 令 <math id="M63"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118793&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118790&type= 23.53733444 4.48733330 ， <math id="M64"><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118798&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118796&type= 24.04533386 4.48733330 ， <math id="M65"><mi>P</mi><mo>+</mo><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118808&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118802&type= 31.24200058 4.48733330 。若 P ≠ Q ，则点加运算公式为 <math display="block" id="M66"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mi>H</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>I</mi><mi>H</mi><mo>+</mo><msup><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>J</mi><mo>+</mo><mi>a</mi><msup><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mi>J</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>C</mi><mi>H</mi><mo>+</mo><mi>F</mi><mi>B</mi><mi>J</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><mi>I</mi><mo>+</mo><mi>H</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mi>I</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118817&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118813&type= 56.38800049 14.98599911 其中： <math id="M67"><mi>A</mi><mo>=</mo><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118824&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118820&type= 12.78466702 3.97933340 ， <math id="M68"><mi>B</mi><mo>=</mo><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118831&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118828&type= 12.36133385 3.97933340 ， <math id="M69"><mi>C</mi><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118840&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118835&type= 14.05466652 3.72533321 ， <math id="M70"><mi>D</mi><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mi>A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118848&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118842&type= 14.81666660 3.72533321 ， <math id="M71"><mi>E</mi><mo>=</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118856&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118852&type= 14.81666660 3.72533321 ， <math id="M72"><mi>F</mi><mo>=</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118864&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118859&type= 15.24000072 3.72533321 ， <math id="M73"><mi>G</mi><mo>=</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118871&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118867&type= 15.57866669 3.72533321 ， <math id="M74"><mi>H</mi><mo>=</mo><mi>F</mi><mi>B</mi><mo>+</mo><mi>G</mi><mi>A</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118878&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118875&type= 23.28333473 2.96333337 ， <math id="M75"><mi>I</mi><mo>=</mo><mi>E</mi><mo stretchy="false">(</mo><mi>C</mi><mo>+</mo><mi>D</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118885&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118882&type= 22.52133369 3.55599999 ， <math id="M76"><mi>J</mi><mo>=</mo><mi>C</mi><mo>+</mo><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118891&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118888&type= 16.59466553 2.96333337 。其倍点运算公式为 <math display="block" id="M77"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mtd></mtr><mtr><mtd><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>C</mi><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>D</mi><mo>+</mo><mi>C</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mi>C</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118898&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118895&type= 48.09066772 14.39333344 其中： <math id="M78"><mi>A</mi><mo>=</mo><msup><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118904&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118901&type= 13.29266644 3.97933340 ， <math id="M79"><mi>B</mi><mo>=</mo><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118911&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118907&type= 12.36133385 3.97933340 ， <math id="M80"><mi>C</mi><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118918&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118915&type= 11.42999935 3.72533321 B ， <math id="M81"><mi>D</mi><mo>=</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118925&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118922&type= 16.08666611 3.72533321 。 根据运算公式，可以看出一次点加运算量为14M+5S，一次倍点运算量为4M+5S。假设 <math id="M82"><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118931&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118928&type= 3.55599999 3.72533321 <math id="M83"><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118939&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118936&type= 6.18066692 2.79399991 时，点加运算量将降到10M+3S。 2　 Co_ Z 点加运算和3 P 运算 1 2.1　射影坐标下的Co_ Z 点加运算 2 令集合 <math id="M84"><msup><mrow><mi>K</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>\</mo><mo>{</mo><mo stretchy="false">(</mo><mn mathvariant="normal">0</mn><mo>:</mo><mn mathvariant="normal">0</mn><mo>:</mo><mn mathvariant="normal">0</mn><mfenced open="" close="}" separators="|"><mrow><mo stretchy="false">)</mo></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118946&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118942&type= 20.48933411 4.31799984 是有限域 GF ( <math id="M85"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118953&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118949&type= 3.38666677 3.04799986 )上的非零三维向量的集合，其上的投影点存在如下等价关系 <math display="block" id="M86"><mo stretchy="false">(</mo><mi>X</mi><mo>:</mo><mi>Y</mi><mo>:</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>=</mo><mo>{</mo><mo stretchy="false">(</mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>c</mi></mrow></msup><mi>X</mi><mo>:</mo><msup><mrow><mi>λ</mi></mrow><mrow><mi>d</mi></mrow></msup><mi>Y</mi><mo>:</mo><mi>λ</mi><mi>Z</mi><mo stretchy="false">)</mo><mo>:</mo><mi>λ</mi><mo>∈</mo><msup><mrow><mi>K</mi></mrow><mrow><mi mathvariant="normal">*</mi></mrow></msup><mo>}</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118961&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118958&type= 53.42466354 3.81000018 通过这一关系，我们可以将椭圆曲线上的两点 <math id="M87"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118967&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118964&type= 23.53733444 4.48733330 , <math id="M88"><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118975&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118971&type= 24.04533386 4.48733330 通过坐标处理，共享 Z 轴，这就是Co_ Z 的主要思路。上节给出了射影坐标下的点加运算公式，当引入Co_ Z 后，可以推导出射影坐标系下改进后的Co_ Z 点加运算公式。 2.1.1　López & Dahab射影坐标下的Co_ Z 点加 3 López & Dahab射影坐标下的两点 P, Q 在仿射坐标下表示为 <math id="M89"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac><mo>,</mo><mfrac><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118983&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118979&type= 23.70666695 8.72066689 ， <math id="M90"><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo>,</mo><mfrac><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118992&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118988&type= 24.12999916 8.72066689 ,分母通分后，两坐标可表示为 <math display="block" id="M91"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo>,</mo><mfrac><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><mi>Z</mi><mmultiscripts><mrow><msup><mrow/><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mprescripts/><mrow><mn mathvariant="normal">1</mn></mrow><mrow/></mmultiscripts><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118999&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4118997&type= 33.61266708 8.88999939 , <math display="block" id="M92"><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><mfrac><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow></mfrac><mo>,</mo><mfrac><mrow><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow><mrow><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mrow></mfrac><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119008&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119004&type= 33.44333649 8.88999939 再转换为射影坐标，则得到 <math display="block" id="M93"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119015&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119012&type= 38.18466568 4.65666676 ， <math display="block" id="M94"><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119023&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119019&type= 37.08399963 4.65666676 因此可将 P , Q 两点转换为 Z 坐标相同的点。 令初始点 <math id="M95"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119031&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119027&type= 23.02933502 4.48733330 ， <math id="M96"><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>:</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119038&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119035&type= 23.45266724 4.48733330 统一 Z 坐标，则 <math id="M97"><mi>P</mi><mo>+</mo><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119047&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119042&type= 31.24200058 4.48733330 的计算公式如下 <math display="block" id="M98"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>C</mi><mo>+</mo><mi>Z</mi><mi>A</mi><mi>C</mi><mo stretchy="false">(</mo><mi>B</mi><mo>+</mo><mi>A</mi><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><mi>a</mi><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mi>Z</mi><mi>A</mi><mi>B</mi><mi>C</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>Z</mi><mi>C</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>C</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>C</mi></mtd></mtr><mtr><mtd><mi>A</mi><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><mi>B</mi><mo>=</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><mi>C</mi><mo>=</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119056&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119051&type= 78.82466125 30.22600174 可将其简化为 <math display="block" id="M99"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>Z</mi><mi>A</mi><mi>B</mi><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>Z</mi><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>Z</mi><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>a</mi><mi>Z</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mi>Z</mi><mi>A</mi><mi>B</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>Z</mi><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mtd></mtr><mtr><mtd><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></mtd></mtr><mtr><mtd><mi>A</mi><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><mi>B</mi><mo>=</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119065&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119060&type= 74.59133148 25.23066711 P+Q 点中 X 、 Y 轴坐标满足 <math id="M100"><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>Z</mi><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>:</mo><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119072&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119069&type= 35.89866638 4.65666676 ～ <math id="M101"><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119080&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119076&type= 15.91733360 4.48733330 ，因此Co_ Z 点加运算的输出点 P+Q 和输入点 P 具有相同的 Z 坐标，而统一 Z 轴变换已经在Co_ Z 点加运算中计算出来，所以不需要额外的运算量。点加算法如算法1。 算法1  López & Dahab射影坐标下Co_ Z 点加算法(LDCo_Z⁃ADD) 输入：Co_ Z <math id="M102"><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo></math> , <math id="M103"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>:</mo><mi>Z</mi><mo stretchy="false">)</mo></math> ， <math id="M104"><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>:</mo><mi>Z</mi><mo stretchy="false">)</mo></math> 输出： <math id="M105"><mi>P</mi><mo>+</mo><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false">)</mo></math> ， <math id="M106"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal">'</mi><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal">'</mi><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false">)</mo></math> <math id="M107"><mi>A</mi><mo>←</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn></mrow></msup></math> ； <math id="M108"><mi>B</mi><mo>←</mo><mi>Z</mi><mi>A</mi></math> ； <math id="M109"><mi>C</mi><mo>←</mo><mi>Z</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></math> ； <math id="M110"><mi>D</mi><mo>←</mo><mi>B</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mi>a</mi><mi>Z</mi><mo stretchy="false">)</mo></math> ； <math id="M111"><msub><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>←</mo><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>C</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>B</mi><mo>+</mo><mi>D</mi></math> ； <math id="M112"><msub><mrow><mi>Y</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>←</mo><mi>C</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>B</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msub><msup><mrow><mi>C</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>A</mi><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub></math> ； <math id="M113"><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>←</mo><mi>Z</mi><mi>B</mi></math> ； <math id="M114"><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal">'</mi><mo>←</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>B</mi></math> ； <math id="M115"><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal">'</mi><mo>←</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>A</mi><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub></math> ； 返回 <math id="M116"><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false">)</mo></math> ， <math id="M117"><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal">'</mi><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal">'</mi><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> 算法1所需运算量为10M+3S，比 <math id="M118"><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119092&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119090&type= 10.41399956 3.72533321 时的点加运算量减少了2S。由于在算法1计算过程中，输入点 P 与输出点 P+Q 的 Z 轴的统一运算量较大，因此在López & Dahab射影坐标下Co_ Z 点加的效率提升不是很明显。 2.1.2　Jacobian射影坐标下的Co_ Z 点加运算 3 令初始点 <math id="M119"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>:</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119097&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119094&type= 48.09066772 4.48733330 ，统一 Z 坐标，则 <math id="M120"><mi>P</mi><mo>+</mo><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119104&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119100&type= 31.24200058 4.48733330 的计算公式可简化为 <math display="block" id="M121"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>Z</mi><mi>A</mi><mi>B</mi><mo>+</mo><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>a</mi><msup><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>B</mi><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">(</mo><mi>Z</mi><mi>A</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mi>Z</mi><mi>A</mi></mtd></mtr><mtr><mtd><mi>A</mi><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr><mtr><mtd><mi>B</mi><mo>=</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119109&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119106&type= 61.72200012 25.23066711 P+Q 点中 X 、 Y 轴坐标满足 <math id="M122"><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><mi>A</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>:</mo><mi>Z</mi><mi>A</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119115&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119112&type= 27.00866890 4.65666676 ～ <math id="M123"><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>:</mo><mi>Z</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119121&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119118&type= 15.91733360 4.48733330 ，因此Co_ Z 点加运算的输出点 P+Q 和输入点 P 具有相同的 Z 坐标，而统一 Z 轴变换已经在Co_ Z 点加运算中计算出来，所以不需要额外的运算量。点加算法如算法2。 算法2  Jacobian射影坐标下Co_ Z 点加算法(JCo_Z⁃ADD) 输入：Co_ Z <math id="M124"><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>Q</mi><mo stretchy="false">)</mo></math> , <math id="M125"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>:</mo><mi>Z</mi><mo stretchy="false">)</mo></math> ， <math id="M126"><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><msub><mrow><mi>Y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>:</mo><mi>Z</mi><mo stretchy="false">)</mo></math> 输出: <math id="M127"><mi>P</mi><mo>+</mo><mi>Q</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false">)</mo></math> ， <math id="M128"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal">'</mi><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal">'</mi><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false">)</mo></math> <math id="M129"><mi>A</mi><mo>←</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn></mrow></msup></math> ； <math id="M130"><mi>B</mi><mo>←</mo><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>2</mn></mrow></msub><msup><mrow><mo stretchy="false">)</mo></mrow><mrow><mn>2</mn></mrow></msup></math> ； <math id="M131"><mi>C</mi><mo>←</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub></math> ； <math id="M132"><mi>D</mi><mo>←</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>A</mi></math> ； <math id="M133"><mi>E</mi><mo>←</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>C</mi></math> ； <math id="M134"><msub><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>←</mo><mi>B</mi><mo>+</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mi>D</mi><mo>+</mo><mi>A</mi><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mi>a</mi><msup><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></math> ； <math id="M135"><msub><mrow><mi>Y</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>←</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><mi>A</mi><mi>E</mi><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">)</mo></math> ； <math id="M136"><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>←</mo><mi>Z</mi><mi>C</mi></math> ； <math id="M137"><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal">'</mi><mo>←</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>A</mi></math> ； <math id="M138"><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal">'</mi><mo>←</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>A</mi><mi>C</mi></math> ； 返回 <math id="M139"><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false">)</mo></math> ， <math id="M140"><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal">'</mi><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mi mathvariant="normal">'</mi><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> Jacobian射影坐标下Co_ Z 点加操作所需运算量为8M+3S，比 <math id="M141"><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119135&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119132&type= 10.41399956 3.72533321 时的点加运算量减少了2M。 2.2　有限域 <math id="M142"><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119142&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119139&type= 11.93799973 3.81000018 上的3 P 运算 2 本小节将计算有限域 <math id="M143"><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119149&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119147&type= 11.93799973 3.81000018 下的3 P 运算，首先回顾一下倍点的运算公式。 令 <math id="M144"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119156&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119153&type= 18.28800011 4.74133348 ， <math id="M145"><mn mathvariant="normal">2</mn><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119161&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119159&type= 20.15066719 4.74133348 ,其中 <math display="block" id="M146"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msup><mrow><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>a</mi></mtd></mtr><mtr><mtd><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>-</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>λ</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mfrac><mrow><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow></mfrac></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119166&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119164&type= 43.09533310 20.31999969 计算 <math id="M147"><mn mathvariant="normal">3</mn><mi>P</mi><mo>=</mo><mn mathvariant="normal">2</mn><mi>P</mi><mo>+</mo><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>,</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119174&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119169&type= 36.49133301 4.74133348 ，Dimitrov等 [ 14 14 ] 给出了仿射坐标系下的3 P 运算公式 <math display="block" id="M148"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mo stretchy="false">(</mo><mfrac><mrow><msup><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>α</mi></mrow></mfrac><mo stretchy="false">)</mo></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mfrac><mrow><msup><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>α</mi></mrow></mfrac><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr><mtr><mtd><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mfrac><mrow><mi>β</mi></mrow><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mi>α</mi></mrow></mfrac><mo stretchy="false">(</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>+</mo><mi>y</mi></mtd></mtr><mtr><mtd><mi>α</mi><mo>=</mo><msup><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>+</mo><msup><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><mi>b</mi></mtd></mtr><mtr><mtd><mi>β</mi><mo>=</mo><mi>α</mi><mo stretchy="false">(</mo><msup><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><msub><mrow><mi>x</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>+</mo><mi>b</mi></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119181&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119178&type= 46.22800064 31.32666779 因此，仿射坐标系下计算一次3 P 运算量为1I+3S+6M，这种算法比文献[ 15 15 ]减少了1M+1S.在Dimitrov提出的3 P 运算公式的基础上，可以推导出射影坐标下的3 P 运算公式。本文对输入点的 Z 坐标进行特殊化处理，能够改进3 P 运算公式，减少运算量。 2.2.1　López & Dahab射影坐标下的3 P 运算 3 2008年，顾海华 [ 16 16 ] 提出一种López & Dahab射影坐标下的3 P 运算公式(TPL)： <math id="M149"><mn mathvariant="normal">3</mn><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119189&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119184&type= 25.39999962 4.48733330 ，则 <math display="block" id="M150"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">(</mo><mi>H</mi><mo>+</mo><msup><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mi>F</mi><mi>H</mi><mi>G</mi><mo>+</mo><mo stretchy="false">(</mo><mi>B</mi><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo stretchy="false">)</mo><msup><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup></mtd></mtr><mtr><mtd><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><msup><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mtd></mtr></mtable></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119195&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119192&type= 42.16399765 14.64733315 其中: <math id="M151"><mi>A</mi><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119200&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119198&type= 11.68400002 3.72533321 <math id="M152"><msup><mrow/><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119206&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119203&type= 1.18533337 3.30199981 ， <math id="M153"><mi>B</mi><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119213&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119210&type= 15.15533447 3.72533321 ， <math id="M154"><mi>C</mi><mo>=</mo><mi>A</mi><mi>B</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119219&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119215&type= 12.78466702 2.79399991 ， <math id="M155"><mi>D</mi><mo>=</mo><mi>A</mi><mo>-</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119224&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119222&type= 19.64266586 3.72533321 ， <math id="M156"><mi>E</mi><mo>=</mo><mi>a</mi><msup><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>-</mo><mi>D</mi><mo stretchy="false">(</mo><mi>D</mi><mo>+</mo><mi>B</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119233&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119229&type= 34.45933533 3.81000018 ， <math id="M157"><mi>F</mi><mo>=</mo><mi>C</mi><mo>+</mo><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119238&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119236&type= 17.35666656 2.96333337 ， <math id="M158"><mi>G</mi><mo>=</mo><mi>B</mi><mi>C</mi><mo>+</mo><mi>F</mi><mi>D</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119246&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119241&type= 22.77533340 2.96333337 ， <math id="M159"><mi>H</mi><mo>=</mo><msup><mrow><mi>B</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mi>E</mi></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119253&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119249&type= 14.39333344 3.04799986 。 这种López & Dahab射影坐标下的3 P 运算量为12M+4S。 2.2.2　Jacobian射影坐标下的3 P 运算 3 Jacobian射影坐标下的 <math id="M160"><mn mathvariant="normal">3</mn><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119258&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119255&type= 25.39999962 4.48733330 ，运算经过简化可以写成 <math display="block" id="M161"><mfenced open="{" close="" separators="|"><mrow><mtable columnalign="left"><mtr><mtd><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mi>a</mi><msup><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>G</mi><mo stretchy="false">(</mo><mi>G</mi><mo>+</mo><mi>F</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></mtd></mtr><mtr><mtd><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mo stretchy="false">(</mo><mi>G</mi><mo>+</mo><mi>F</mi><mo stretchy="false">)</mo><mi>H</mi><mo>+</mo><msup><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo stretchy="false">(</mo><mi>G</mi><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mi>E</mi><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo stretchy="false">)</mo></mtd></mtr><mtr><mtd><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msub><mo>=</mo><mi>E</mi><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mtd></mtr></mtable></mrow></mfenced></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119265&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119261&type= 57.57333374 14.98599911 其中： <math id="M162"><mi>E</mi><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119270&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119267&type= 25.23066711 4.06400013 ， <math id="M163"><mi>F</mi><mo>=</mo><mi>E</mi><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119275&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119273&type= 13.46199894 3.72533321 ， <math id="M164"><mi>G</mi><mo>=</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119282&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119278&type= 31.83466721 4.06400013 ， <math id="M165"><mi>H</mi><mo>=</mo><mi>a</mi><msup><mrow><mi>F</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><mi>G</mi><mo stretchy="false">(</mo><mi>G</mi><mo>+</mo><mi>F</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>E</mi></mrow><mrow><mn mathvariant="normal">3</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119286&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119284&type= 43.18000031 3.81000018 ， <math id="M166"><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msup><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><mo>+</mo><mi>b</mi><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">8</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119292&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119289&type= 25.82333374 4.06400013 ， <math id="M167"><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msup><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">4</mn></mrow></msup><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>+</mo><mo stretchy="false">(</mo><msup><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>+</mo><mi>Z</mi><mmultiscripts><mrow><mo stretchy="false">)</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub></mrow><mprescripts/><mrow><mn mathvariant="normal">2</mn></mrow><mrow/></mmultiscripts></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119298&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119295&type= 58.50466537 4.65666676 ， <math id="M168"><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msub><mo>=</mo><msub><mrow><mi>X</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><msup><mrow><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub></mrow><mrow><mn mathvariant="normal">2</mn></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119305&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119302&type= 17.69533348 3.97933340 。 因此，此3 P 运算量为13M+7S。 对 Z 坐标进行特殊化处理，当 <math id="M169"><msub><mrow><mi>Z</mi></mrow><mrow><mn mathvariant="normal">1</mn></mrow></msub><mo>=</mo><mn mathvariant="normal">1</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119311&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119308&type= 10.41399956 3.72533321 时，可以得出改进后的3 P 运算公式。通过分析可知，改进后的3 P 运算代价为12M+3S，具体算法如算法3。 算法3  Jacobian射影坐标下新3 P 算法(New⁃TPL) 输入： <math id="M170"><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>:</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∈</mo><mi>E</mi><mo stretchy="false">(</mo><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo></math> 输出： <math id="M171"><mn>3</mn><mi>P</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false">)</mo></math> <math id="M172"><mi>A</mi><mo>←</mo><msup><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup></math> ； <math id="M173"><mi>B</mi><mo>←</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub></math> ； <math id="M174"><mi>C</mi><mo>←</mo><msup><mrow><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mrow><mn>3</mn></mrow></msup></math> ； <math id="M175"><mi>D</mi><mo>←</mo><mi>C</mi><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub></math> ； <math id="M176"><mi>E</mi><mo>←</mo><mi>A</mi><mi>B</mi></math> ； <math id="M177"><mi>F</mi><mo>←</mo><mi>E</mi><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>2</mn></mrow></msub></math> ； <math id="M178"><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>←</mo><msup><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><mi>b</mi></math> ； <math id="M179"><msub><mrow><mi>Y</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>←</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><msup><mrow><mi>A</mi></mrow><mrow><mn>2</mn></mrow></msup><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>2</mn></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo>+</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mi>X</mi><mmultiscripts><mrow><mo stretchy="false">)</mo></mrow><mprescripts/><mrow><mn>1</mn></mrow><mrow/></mmultiscripts></math> ； <math id="M180"><msub><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>←</mo><mi>F</mi><mo stretchy="false">(</mo><mi>F</mi><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>D</mi><mo stretchy="false">)</mo><mo>+</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>D</mi><mo>+</mo><mi>a</mi><mi>A</mi><mo stretchy="false">)</mo></math> ； <math id="M181"><msub><mrow><mi>Y</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>←</mo><msup><mrow><mi>D</mi></mrow><mrow><mn>2</mn></mrow></msup><mo stretchy="false">(</mo><mi>C</mi><mi>F</mi><mo>+</mo><mi>E</mi><mi>D</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>D</mi><mo>+</mo><mi>F</mi><mo stretchy="false">)</mo></math> ； <math id="M182"><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>←</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>1</mn></mrow></msub><mi>D</mi></math> ； 返回 <math id="M183"><mo stretchy="false">(</mo><msub><mrow><mi>X</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Y</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>:</mo><msub><mrow><mi>Z</mi></mrow><mrow><mn>3</mn></mrow></msub><mo stretchy="false">)</mo></math> 2.3　Co_ Z 对称三进制标量乘算法 2 标量 k 可展开成对称三进制形式(symmetric ternary form，STF) [ 17 17 ] ，相对二进制形式，它能够缩短展开后的链长，减少点加和3 P 运算次数。将对称三进制应用到标量乘运算中，具体的标量乘算法可参考文献[ 3 3 ].在对称三进制标量乘算法中，每轮迭代中只有当 <math id="M184"><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>≠</mo><mn mathvariant="normal">0</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119323&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119320&type= 10.07533360 3.97933340 时，才执行点加运算，而根据对称三进制的展开式，非零数出现的概率为 <math id="M185"><mn mathvariant="normal">2</mn><mo>/</mo><mn mathvariant="normal">3</mn></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119330&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119327&type= 5.58799982 3.30199981 ，因此其运算代价为 <math id="M186"><mi>n</mi><mo stretchy="false">(</mo><mi>T</mi><mo>+</mo><mn mathvariant="normal">2</mn><mi>A</mi><mo>/</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119335&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119332&type= 20.99733353 3.72533321 ,其中 A 表示点加的运算代价， T 表示3 P 的运算代价。 为提高标量乘的安全性，使其能够抵御简单能量攻击(simple power analysis,SPA)，通过增加标量乘算法的运算量，文献[ 18 18 ]提出了一种可抵御SPA攻击的对称三进制标量乘算法。将本文提出的点加和3 P 运算公式与文献[ 18 18 ]算法相结合，可优化标量乘算法的效率，如算法4。 算法4  Co_ Z 对称三进制标量乘算法 输入： <math id="M187"><mi>P</mi><mo>,</mo><mi mathvariant="normal">S</mi><mi mathvariant="normal">T</mi><mi mathvariant="normal">P</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></math> 输出： <math id="M188"><mi>k</mi><mi>P</mi></math> <math id="M189"><msub><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">∞</mi></math> ， <math id="M190"><msub><mrow><mi>P</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>=</mo><mi mathvariant="normal">∞</mi></math> ， <math id="M191"><msub><mrow><mi>P</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mi>P</mi></math> ， <math id="M192"><msub><mrow><mi>P</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msub><mo>=</mo><mo>-</mo><mi>P</mi></math> ； <math id="M193"><mi mathvariant="normal">f</mi><mi mathvariant="normal">o</mi><mi mathvariant="normal">r</mi><mi>i</mi><mo>=</mo><mi>n</mi><mo>-</mo><mn>1</mn><mi mathvariant="normal">t</mi><mi mathvariant="normal">o</mi><mn>0</mn><mo>,</mo><mi>i</mi><mo>-</mo><mo>-</mo></math> <math id="M194"><msub><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>←</mo><mi mathvariant="normal">N</mi><mi mathvariant="normal">e</mi><mi mathvariant="normal">w</mi><mo>-</mo><mi mathvariant="normal">T</mi><mi mathvariant="normal">P</mi><mi mathvariant="normal">L</mi><mo stretchy="false">(</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo stretchy="false">)</mo></math> ； <math id="M195"><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>←</mo></math> Co_ Z ( <math id="M196"><msub><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>,</mo><msub><mrow><mi>P</mi></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math> )； <math id="M197"><msub><mrow><mi>Q</mi></mrow><mrow><mo>-</mo><mn>1</mn></mrow></msub><mo>←</mo><msub><mrow><mi>Q</mi></mrow><mrow><mn>1</mn></mrow></msub></math> ； <math id="M198"><msub><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>←</mo><msub><mrow><mi>Q</mi></mrow><mrow><msub><mrow><mi>k</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></msub></math> ； end for 返回 <math id="M199"><msub><mrow><mi>Q</mi></mrow><mrow><mn>0</mn></mrow></msub></math> 在算法4中，每轮循环都需要执行一次Co_ Z 点加和一次3 P 运算，因此整个标量乘算法的运算量为 n ( A + T )。文献[ 18 18 ]中的可抵御SPA攻击的对称三进制标量乘算法采用的是仿射坐标系，每轮循环运算量为2I+5S+9M，这里求逆与域乘的运算量之比为8。由于在算法4中引入了Co_ Z 点加算法，减少了每轮循环中点加和3 P 运算所需的M、S计算次数，因此提高了运算效率。 3　 运算效率分析 1 将本文提出的Co_ Z 点加、3 P 运算公式应用到文献[ 3 3 ]的对称三进制标量乘算法中，比较不同运算公式下的标量乘算法运算效率（见 表1 表1 ）.160⁃bit对称三进制标量乘算法的运算量为 <math id="M200"><mn mathvariant="normal">101</mn><mo stretchy="false">(</mo><mi>T</mi><mo>+</mo><mn mathvariant="normal">2</mn><mi>A</mi><mo>/</mo><mn mathvariant="normal">3</mn><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119349&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119346&type= 24.46866608 3.72533321 。 2 坐标系 2 点加（ A ） 3 P ( T ) 标量乘 Lopez & Dahab射影坐标 2 Co_ Z 10M+3S 12M+4S 1885M+606S 2 Jacobian射影坐标 2 <math id="M204"><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>1</mn></math> 10M+3S 13M+7S 1986M+909S 2 Co_ Z 8M+3S 12M+3S 1751M+505S 在有限域 <math id="M201"><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119359&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119357&type= 11.93799973 3.81000018 内，平方运算更加高效，因此在二元域内，可以忽略计算平方(S)的时间 [ 12 12 ] ，只比较计算域乘所需要的时间代价。Jacobian射影坐标下，当Co_ Z 点加与3 P 运算应用到对称三进制标量乘算法中，算法的运算效率提高了12%。若在Jacobian射影坐标和López & Dahab射影坐标下均使用Co_ Z 点加运算公式，则对称三进制标量乘算法在Jacobian射影坐标的运算效率要比López & Dahab射影坐标下提高8%。 在160⁃bit下，对称三进制标量乘算法的运算量为 <math id="M202"><mn mathvariant="normal">101</mn><mo stretchy="false">(</mo><mi>T</mi><mo>+</mo><mi>A</mi><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119367&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119363&type= 18.88066673 3.55599999 。 表2 表2 给出了使用不同点加、3 P 运算公式时，可抵御SPA攻击对称三进制标量乘算法的运算量。在Jacobian射影坐标下，改进后的点加、3 P 运算公式后得到的运算量为2020M+606S，此即为算法4的运算量。 注： a为将本文数据代入文献[ 18 18 ]算法结果；b为本文算法4 2 坐标系 点加（ A ） 3 P ( T ) 标量乘 Lopez & Dahab射影坐标 Co_ Z 10M+3S 12M+4S 2222M+707S a 2 Jacobian射影坐标 <math id="M205"><msub><mrow><mi>Z</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>=</mo><mn>1</mn></math> 10M+3S 13M+7S 2323M+1010S a Co_ Z 8M+3S 12M+3S 2020M+606S b 当引用Co_ Z 操作后，López & Dahab射影坐标下的点加运算量由10M+5S降至10M+3S；Jacobian射影坐标下的点加运算量由10M+3S降至8M+3S。新提出的Jacobian射影坐标下3P运算公式比之前的少了1M+4S。由 表2 表2 可知，Jacobian射影坐标下，结合了Co_ Z 点加和改进后的3 P 运算的算法4的运算效率，相比使用改进前的点加、3 P 运算公式的可抵御SPA攻击对称三进制标量乘算法提高了13%(只比较计算域乘所需要的时间代价)。 4　 结 论 1 本文对有限域 <math id="M203"><mi>G</mi><mi>F</mi><mo stretchy="false">(</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup><mo stretchy="false">)</mo></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119387&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119382&type= 11.93799973 3.81000018 内的点加、3 P 运算公式进行改进，使得运算过程中的域乘、域平方计算次数得到减少。当引入Co_ Z 思想后，通过统一输入点的 Z 坐标， López & Dahab射影坐标下Co_ Z 点加运算量降低了2S；Jacobian射影坐标下Co_ Z 点加运算量降低了2M。此外，当输入点的 Z 轴特殊化处理为1后，Jacobian射影坐标下3 P 运算量减少了1M+4S，都有不同程度的效率提高。将提出的Co_ Z 点加、3 P 运算公式与可抵御SPA攻击的标量乘算法相结合，提出一种Co_ Z 对称三进制标量乘算法，使得标量乘算法的运算效率提高了13%。下一步，可选取合适的椭圆曲线，将Co_ Z 点加与曲线相结合，提出更高效的标量乘算法，使其能应用到具体的硬件环境中。 参考文献 MILLER V S . 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A high⁃performance elliptic curve cryptographic processor overGF(p) with SPA resistance FENG X , LI S G . A high⁃speed and SPA⁃resistant implementation of ECC point multiplication over GF( p ) [C]// 2017 IEEE Trustcom/BigDataSE/ICESS . Washington D C : IEEE , 2017 : 255 ⁃ 260 .DOI:10.1109/Trustcom/BigDataSE/ICESS.2017.245 A high⁃speed and SPA⁃resistant implementation of ECC point multiplication over GF(p) MELONI N . New point addition formulae for ECC applications [C]// WAIFI 07 Proceedings of the 1st International Workshop on Arithmetic of Finite Fields . Berlin : Springer⁃Verlag . 2007 : 189 ⁃ 201 . DOI:10.1007/978⁃3⁃540⁃73074⁃3_15. New point addition formulae for ECC applications GOUNDAR R R , JOYE M , MIYAJI A . Co⁃ Z addition formulæ and binary ladders on elliptic curves [C]// Cryptographic Hardware and Embedded Systems. Berlin: Springer⁃Verlag . 2010 : 65 ⁃ 79 . DOI: 10.1007/978⁃3⁃642⁃15031⁃9_5 Co⁃Z addition formulæ and binary ladders on elliptic curves WEI J Z , LIU X L , GUO W . A low⁃time⁃complexity and secure dual⁃field scalar multiplication based on Co⁃ Z protected NAF [J]. IEICE Electron Express , 2014 , 11 ( 11 ): 361 . DOI: 10.1587/elex.11.20140361 http://dx.doi.org/10.1587/elex.11.20140361 . A low⁃time⁃complexity and secure dual⁃field scalar multiplication based on Co⁃Z protected NAF 于伟 , 李宝 , 王鲲鹏 , 等 . 特征3有限域上椭圆曲线的Co⁃ Z Montgomery算法 [J]. 计算机学报 , 2017 , 40 ( 5 ): 1121 ⁃ 1133 . 特征3有限域上椭圆曲线的Co⁃Z Montgomery算法 YU W , LI B , WANG K P , et al . Co⁃ Z Montgomery algorithm on elliptic curves over finite fields of characteristic three [J]. Chinese Journal of Computers , 2017 , 40 ( 5 ): 1121 ⁃ 1133 (Ch) . Co⁃ZMontgomery algorithm on elliptic curves over finite fields of characteristic three 李俊芳 , 崔建双 . 抗MOV规约法攻击的一类安全椭圆曲线 [J]. 计算机工程与应用 , 2004 ( 36 ): 67 ⁃ 68+101 . 抗MOV规约法攻击的一类安全椭圆曲线 LI J F , CUI J S . A Method to generate elliptic curves [J]. 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GF ( <math id="M206"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119395&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119391&type= 2.45533323 2.03200006 )椭圆曲线上的Co_ Z 点加算法[J].武汉大学学报（理学版）,2019,65(2):207⁃212. 刘双根,丁媛媛,施瑞, 等.GF(2m)椭圆曲线上的Co_Z点加算法[J].武汉大学学报（理学版）,2019,65(2):207⁃212. LIU Shuanggen,DING Yuanyuan,SHI Rui, et al .Co_ Z Addition on Elliptic Curves over Finite Fields GF ( <math id="M207"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>m</mi></mrow></msup></math> http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119405&type= http://html.publish.founderss.cn/rc-pub/api/common/picture?pictureId=4119401&type= 2.45533323 2.03200006 ) [J]. J Wuhan Univ （ Nat Sci Ed ）,2019,65(2):207⁃212. DOI:10.14188/j.1671⁃8836.2019.02.010(Ch). LIU Shuanggen,DING Yuanyuan,SHI Rui,et al.Co_Z Addition on Elliptic Curves over Finite Fields GF(2m) [J].J Wuhan Univ（Nat Sci Ed）,2019,65(2):207⁃212. DOI:10.14188/j.1671⁃8836.2019.02.010(Ch). 国家自然科学基金(61872292)；陕西省自然科学基金(2017JQ6010) 2018-07-02 2019-04-24 2020-11-25 武汉大学学报（理学版） 2 65