CN1946020A - 安全高效率的椭圆曲线加解密参数 - Google Patents

安全高效率的椭圆曲线加解密参数 Download PDF

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Publication number
CN1946020A
CN1946020A CN 200610142296 CN200610142296A CN1946020A CN 1946020 A CN1946020 A CN 1946020A CN 200610142296 CN200610142296 CN 200610142296 CN 200610142296 A CN200610142296 A CN 200610142296A CN 1946020 A CN1946020 A CN 1946020A
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Prior art keywords
elliptic curve
parameter
decryption
encryption
encrypting
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Inventor
陈建华
汪朝晖
胡进
胡志金
孙金龙
张家宏
阳凌怡
张丽娜
何德彪
汪玉
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BEIJING HUADA INFOSEC TECHNOLOGY Ltd
Wuhan University WHU
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BEIJING HUADA INFOSEC TECHNOLOGY Ltd
Wuhan University WHU
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Abstract

本发明涉及椭圆曲线加解密方法。本发明的椭圆曲线加解密方法中，椭圆曲线加解密基域为素域Fp，其中p为224位，并且p满足
Figure 200610142296.6_AB_0
，ki∈{－1，0，1}或p满足p＝2224－2k－1，1≤k≤223。本发明的目的是提出一种针对224位的椭圆曲线提出一类ECC参数的选择方法，使得实现安全高效率的ECC软件和硬件更为可行。

Description

安全高效率的椭圆曲线加解密参数
技术领域
本发明涉及数据加解密，是影响数据加解密安全和效率的系统参数。
背景技术
椭圆曲线密码系统(Elliptic Curve Cryptosystems，简称ECC)自1985年由Neal Koblitz和Victor Miller提出以来，由于其相对于RSA的全方面的优势(更强的安全性、更高的实现效率、更省的实现代价)，吸引了大批密码学工作者就其安全性和实现方法作了大量的研究，并已逐渐被国际各大标准组织采纳做为公钥密码标准(IEEE P1363、ANSI X9、ISO/IEC、和IETF等)，成为主流应用的公钥密码之一。
ECC的安全性和实现性能，在很大程度上依赖于椭圆曲线参数的选取，包括基域的选取和基域上椭圆曲线方程的选取。
在ECC应用中，一般基域选择为二元扩域F2m或素域Fp(p为大于3的素数)：当选择基域为F2m时，通过选择F2m中的模多项式为三项式或五项式以及用高斯正规基表示F2m中的元素等技术手段，可以很大程度上提高F2m上各种算术运算的性能，从而提高ECC的性能；当选择基域为Fp时，通过选择p为特殊的素数可以很大程度上提高Fp上各种算术运算的性能。
在专利“密码体系中公钥交换的方法和装置”(“Method andapparatus for public key exchange in a cryptographic system”，专利号：US5,159,632)中给出了一类特殊的素数，该类素数型如：2k-c其中0＜c＜232。针对该类素数有快速取模的方法，从而可以提高Fp上各种算术运算的性能。
一般Fp上的椭圆曲线具有Weiestrss方程：y2＝x3+ax+b，a、b∈Fp，选择特殊的a可以提高椭圆曲线点乘的运算效率。
椭圆曲线方程的选取决定了椭圆曲线的阶(即椭圆曲线上点的个数，记为n)，椭圆曲线的阶在ECC应用中扮演着至关重要的作用。在ECC应用中，出于安全性的考虑一般要求n含有大素数因子，最佳选择是当基域为Fp时，n本身就是大素数。当椭圆曲线的阶n为大素数时，椭圆曲线群的基点G＝(Gx，Gy)可以是椭圆曲线群上任意一点。
发明内容
本发明的目的是针对224位的椭圆曲线提出一种参数的选择方法及基于该方法的224位ECC参数，使得实现安全高效率的ECC软件和硬件更为可行。
根据本发明的一个方面，提供一种椭圆曲线加解密方法，椭圆曲线加解密基域为素域Fp，其中p为224位，并且p满足 p = 2 224 + Σ i = 0 6 2 32 i k i , k i ∈ { - 1,0,1 } . 或p满足p＝2224-2k-1，1≤k≤223。
根据本发明的一个实施例，在素域Fp上的椭圆曲线具有Weierstrass方程：y2＝x3+ax+b，a、b∈Fp，其中，选择a≡-3modp，并且a和b满足(4a3+27b2)modp≠0，同时a和b的选取使得椭圆曲线的阶n为素数。
本发明的另外的实施例中给出了多组p、a和b优选的参数。
附图说明
图1示出采用本发明的加解密参数选择装置的加解密系统的方框图。
具体实施方式
一种椭圆曲线加解密方法，椭圆曲线加解密基域为素域Fp，其中p为224位。本发明具体包括以下两部分的内容：
1.参数选择方法
1.1 224位素域Fp上椭圆曲线参数选择方法
为满足所选参数使得Fp上椭圆曲线系统高效，本发明选取的p为特殊的素数。本发明选取的特殊素数分为两类：一类p满足 p = 2 224 + Σ i = 0 6 2 32 i k i , ki∈{-1，0，1}；另一类p满足p＝2224-2k-1，1≤k≤223。p所具备的特点使得对p取模运算可以通过为数不多的加法和减法操作完成，从而可以设计高效的模乘软件和硬件实现算法。
为提高椭圆曲线点乘效率，本发明选取的椭圆曲线方程参数a满足a≡-3modp。a和b应满足(4a3+27b2)modp≠0同时为满足安全性本发明选取的a和b使得椭圆曲线的阶n为素数。
2.安全高效率的椭圆曲线系统参数
在满足上述参数选择方法的条件下，p、a和b进一步采用以下各组参数之一。本发明的素数均通过确定性素性检测。为保证椭圆曲线系统安全性，本发明选取的n和p通过MOV判断条件。
本发明中以下的参数均为十六进制表示。
根据参数的选择方法，224位素数p的选择为
1)
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
p＝FFFFFFFFFFFFFFFEFFFFFFFF0000000100000000FFFFFFFEFFFFFFFF
p＝FFFFFFFF000000000000000000000000000000000000000100000001
p＝FFFFFFFF0000000000000001000000000000000000000000FFFFFFFF
p＝FFFFFFFF000000010000000100000001000000000000000000000001
p＝FFFFFFFF000000010000000100000000FFFFFFFEFFFFFFFF00000001
p＝FFFFFFFEFFFFFFFF000000000000000000000000FFFFFFFFFFFFFFFF
p＝FFFFFFFEFFFFFFFF00000000FFFFFFFEFFFFFFFF00000000FFFFFFFF
p＝FFFFFFFEFFFFFFFEFFFFFFFF00000000000000010000000100000001
p＝FFFFFFFEFFFFFFFEFFFFFFFEFFFFFFFF00000000FFFFFFFF00000001
p＝FFFFFFFEFFFFFFFEFFFFFFFEFFFFFFFEFFFFFFFF0000000100000001
2)
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
椭圆曲线参数选择为以下各组中的任意一个。
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝DCB4307F23CA5F944E9F23CF61A0D1F6FD3708A87B7AE976716E68B
Gx＝1011D487D3D6F256A3FBF97F32CACE82DA9D26EE6BBCA871EE045DF8
Gy＝EAB80678DB3B1929B4A779CBBBD5543667C9FE9915847A8CC13093CF
n＝FFFFFFFFFFFFFFFEFFFFFFFEFFFF3DED44B6BC3AF1E7FED549DB388B
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝35F23F53A51F4D9BB689A7071A9218EA426948D22FC1CCBABF59F70
Gx＝1608C085BA6DC7ECEA756615C488ED797BC9C4FBC9118CAD8E4A6415
Gy＝5B0646D2BB213BE1FCFE9801CC89983FD89DD0D3E22F44F33CF688ED
n＝FFFFFFFFFFFFFFFEFFFFFFFEFFFF58602970D61D570A6FA14E32DDEB
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝67DD503F0045DC305551E3C071C757D9D189352DB4FBE1C0F52444B2
Gx＝2FDA3CFE677DB3F91B08C6D3B45652309C9B460DB629BE741F9F6B3
Gy＝1BD0D2FA2991917F809663F7D6A6557EDF933007E67615C35F8BDED3
n＝FFFFFFFFFFFFFFFEFFFFFFFEFFFE579C1262BBBD93EE592F2D864B45
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝B4C9085AE39EF3E1638A2540D0B367B7FBAC9FDF3908DD8AD0646FEB
Gx＝22648F44B91CB42E444FDF1B07AC0EF529FE45F9661EF4A1FABEC067
Gy＝2DE2F870C74D356FE415DB1920E601C164895C46FF677E26B09D3925
n＝FFFFFFFFFFFFFFFEFFFFFFFEFFFE2DEA1667F52001AC41EC4A4385F1
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝1248940303D72DE541E65E152131EDB4DD0728CF10EA33F4F07AA8F9
Gx＝4B2C9376020B9DE449768FA87E9FA0862DF0BBD34A36FDD4D4477373
Gy＝6DCB7718C64642231DF10C935FD7E8BCBCF23C87CDC1BFD857DCDDEF
n＝FFFFFFFFFFFFFFFEFFFFFFFF0001096006D4A812714381686D1BCD29
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝EDE90BC2317ACEA4FAC05394C10FF1972748E806C8CB2F151893E97C
Gx＝3E115B26D2299312B5E149177CD87313BBEB68FC6529FE56DAF4C347
Gy＝76C24DB215839F4890FD3B5B6ED3B0FFD0443206771BE3359432AFA3
n＝FFFFFFFEFFFFFFFFFFFFFFFFFFFFC470EE78E46462D1559CBAB1684B
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝18DFDB909315F8050B291DBFD6D79EE347B125BF815C3D43A9D53FDA
Gx＝1B66919C660D91263C3EDA3BE43C40F8F7470CD386DA901A30B36EC8
Gy＝136F80185FCFF03C6D204BDC2AAA97B93BD7534DEDDEB56B0BE98F9D
n＝FFFFFFFEFFFFFFFFFFFFFFFFFFFE16E4C6603718C374BC2C403A3E3D
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝65D470FFA6063282AC80670FD970A057472F844D5288548203C10B14
Gx＝7597861368B6BD2D15BBB4D44B5EFF2F26EC5F24D495224E7F163FE
Gy＝247D42457323B85C22C94B9BF497739BF0135F66A9BB5BAA4AADFCB6
n＝FFFFFFFEFFFFFFFFFFFFFFFFFFFFB09B25199C3C45A15F7145FDAC0D
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝6239851169AF336253D6B269966CABF07DE36B941D4EDE1D578454F3
Gx＝7E8342386104A93D329064F9F168F65A8D4AC7438CA347A79C1C2748
Gy＝CA8FEC052F6C7101DD760C4DFF9F8DFBC6BB7CC420A3E06627D156B9
n＝FFFFFFFEFFFFFFFFFFFFFFFFFFFF855D60E9C373CC8FE23AA6FC336B
1)
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝2700FF77B1235CAF22A35F20D78AB7B04D12FF4006DD9FC6525840B6
Gx＝7A4F3469E3E07967BA4C1729C86EB4AD3071612965CF88A094C9184C
Gy＝137E03C2068E68721CE71A000D88C0070B9BFD71EF18B027E4A79ACB
n＝FFFFFFFFFFFFFFFFFFFFFFFFFFFEE107B67D487E68FB6E672C2DB031
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝67E0F23C36BDA999E8D7AB3A9EF31BAB11B19C247901C7F402720015
Gx＝615FBFCC9C14291BEA3EA547DC3FE26DA0D2EB427A5436758E9686E2
Gy＝1B51AB3ED9536850335F268E240C8BFFF691C4421058C2358C9B75DF
n＝1000000000000000000000000000113AE63259EE823A1360A1934BCFF
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝9A88C3406278CBCBA1688FFA6C4371BE472213999038393A2369E1C2
Gx＝18A7FC17C22CB358FBDFE391CF0749DE8CDB4E6DE7B5E0BD75F874AA
Gy＝10AA2BF80C9993AD57AFA479CB3DADA382F76742554B4C8E471A1980
n＝FFFFFFFFFFFFFFFFFFFFFFFFFFFF32D37DA7EFB23D7840C0E7675041
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝8B22A7551C6ED668A52E767676F5FD390452940FA5F24221E9654686
Gx＝7BCD48550A57A5ABC01B99177E3E292B22B8051B5FCA9A302AE80D45
Gy＝919DF9F61BAC799180D38E5879BFDCC8ABB7179597DAE4B036585945
n＝100000000000000000000000000001C3620B759F9F761AAFD3DD33B89
图1示出采用本发明的加解密参数选择装置的加解密系统的方框图。加解密系统一般包含加密装置110、解密装置120、参数选择器130和密钥生成装置140。参数选择器130用于选择椭圆曲线加解密参数，可以由专门提供安全加密算法的供应商提供，也可能是加密装置110自己生成。如果加密装置110自己生成加解密参数，则应当与解密装置120选择的参数相同。本发明的参数选择器130按照如下方式选择椭圆曲线加解密参数：
椭圆曲线加解密基域为素域Fp，其中p为224位。本发明具体包括以下两部分的内容：
1.参数选择方法
1.1 224位素域Fp上椭圆曲线参数选择方法
为满足所选参数使得Fp上椭圆曲线系统高效，本发明选取的p为特殊的素数。本发明选取的特殊素数分为两类：一类p满足 p = 2 224 + Σ i = 0 6 2 32 i k i , ki∈-{-1，0，1}；另一类p满足p＝2224-2k-1，1≤k≤223。p所具备的特点使得对p取模运算可以通过为数不多的加法和减法操作完成，从而可以设计高效的模乘软件和硬件实现算法。
为提高椭圆曲线点乘效率，本发明选取的椭圆曲线方程参数a满足a≡-3modp。a和b应满足(4a3+27b2)modp≠0同时为满足安全性本发明选取的a和b使得椭圆曲线的阶n为素数。
其中加解密参数p、a、b和n的进一步选择可以参考以上对加解密方法的说明，在此不再重复。
参数获取或生成装置111生成加解密参数或从参数选择器130接收到加解密参数后，由公钥获取装置112接收解密装置120的公钥Y。其中，公钥Y由密钥生成装置140生成。加密器113用公钥Y和有参数获取或生成装置111提供的加解密参数对明文m加密，并将密文c发送给加密装置120。
解密装置120的密文接收装置122接收密文c后，解密器123用密钥生成装置140生成的私钥k和参数获取或生成装置121提供的加解密参数对密文c进行解密。

Claims (10)

1.一种椭圆曲线加解密方法，椭圆曲线加解密基域为素域Fp，其中p为224位，并且p满足 p = 2 224 + Σ i = 0 6 2 32 i k i , ki∈{-1，0，1}或p满足p＝2224-2k-1，1≤k≤223。
2.如权利要求1的椭圆曲线加解密方法，在素域Fp上的椭圆曲线具有Weierstrass方程：y2＝x3+ax+b，a、b∈Fp，其中，选择a≡-3mod p，并且a和b满足(4a3+27b2)mod p≠0，同时a和b的选取使得椭圆曲线的阶n为素数。
3.如权利要求1的椭圆曲线加解密方法，素数p的选择为
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
p＝FFFFFFFFFFFFFFFEFFFFFFFF0000000100000000FFFFFFFEFFFFFFFF
p＝FFFFFFFF000000000000000000000000000000000000000100000001
p＝FFFFFFFF0000000000000001000000000000000000000000FFFFFFFF
p＝FFFFFFFF000000010000000100000001000000000000000000000001
p＝FFFFFFFF000000010000000100000000FFFFFFFEFFFFFFFF00000001
p＝FFFFFFFEFFFFFFFF000000000000000000000000FFFFFFFFFFFFFFFF
p＝FFFFFFFEFFFFFFFF00000000FFFFFFFEFFFFFFFF00000000FFFFFFFF
p＝FFFFFFFEFFFFFFFEFFFFFFFF00000000000000010000000100000001
p＝FFFFFFFEFFFFFFFEFFFFFFFEFFFFFFFF00000000FFFFFFFF00000001
p＝FFFFFFFEFFFFFFFEFFFFFFFEFFFFFFFEFFFFFFFF0000000100000001
p＝FFFFFFFFFFFFFF  FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF。
4.如权利要求2的椭圆曲线加解密方法，椭圆曲线参数p、a和b的选择为以下组中的任意一个：
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝DCB4307F23CA5F944E9F23CF61A0D1F6FD3708A87B7AE976716E68B
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝35F23F53A51F4D9BB689A7071A9218EA426948D22FC1CCBABF59F70
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝67DD503F0045DC305551E3C071C757D9D189352DB4FBE1C0F52444B2
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝B4C9085AE39EF3E1638A2540D0B367B7FBAC9FDF3908DD8AD0646FEB
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝1248940303D72DE541E65E152131EDB4DD0728CF10EA33F4F07AA8F9
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝EDE90BC2317ACEA4FAC05394C10FF1972748E806C8CB2F151893E97C
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝18DFDB909315F8050B291DBFD6D79EE347B125BF815C3D43A9D53FDA
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝65D470FFA6063282AC80670FD970A057472F844D5288548203C10B14
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝6239851169AF336253D6B269966CABF07DE36B941D4EDE1D578454F3
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝2700FF77B1235CAF22A35F20D78AB7B04D12FF4006DD9FC6525840B6
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝67E0F23C36BDA999E8D7AB3A9EF31BAB11B19C247901C7F402720015
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝9A88C3406278CBCBA1688FFA6C4371BE472213999038393A2369E1C2
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝8B22A7551C6ED668A52E767676F5FD390452940FA5F24221E9654686。
5.如权利要求4的椭圆曲线加解密方法，其中各组中椭圆曲线的阶n和基点G＝(Gx，Gy)分别为：
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝DCB4307F23CA5F944E9F23CF61A0D1F6FD3708A87B7AE976716E68B
Gx＝1011D487D3D6F256A3FBF97F32CACE82DA9D26EE6BBCA871EE045DF8
Gy＝EAB80678DB3B1929B4A779CBBBD5543667C9FE9915847A8CC13093CF
n＝FFFFFFFFFFFFFFFEFFFFFFFEFFFF3DED44B6BC3AF1E7FED549DB388B
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝35F23F53A51F4D9BB689A7071A9218EA426948D22FC1CCBABF59F70
Gx＝1608C085BA6DC7ECEA756615C488ED797BC9C4FBC9118CAD8E4A6415
Gy＝5B0646D2BB213BE1FCFE9801CC89983FD89DD0D3E22F44F33CF688ED
n＝FFFFFFFFFFFFFFFEFFFFFFFEFFFF58602970D61D570A6FA14E32DDEB
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝67DD503F0045DC305551E3C071C757D9D189352DB4FBE1C0F52444B2
Gx＝2FDA3CFE677DB3F91B08C6D3B45652309C9B460DB629BE741F9F6B3
Gy＝1BD0D2FA2991917F809663F7D6A6557EDF933007E67615C35F8BDED3
n＝FFFFFFFFFFFFFFFEFFFFFFFEFFFE579C1262BBBD93EE592F2D864B45
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝B4C9085AE39EF3E1638A2540D0B367B7FBAC9FDF3908DD8AD0646FEB
Gx＝22648F44B91CB42E444FDF1B07AC0EF529FE45F9661EF4A1FABEC067
Gy＝2DE2F870C74D356FE415DB1920E601C164895C46FF677E26B09D3925
n＝FFFFFFFFFFFFFFFEFFFFFFFEFFFE2DEA1667F52001AC41EC4A4385F1
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝1248940303D72DE541E65E152131EDB4DD0728CF10EA33F4F07AA8F9
Gx＝4B2C9376020B9DE449768FA87E9FA0862DF0BBD34A36FDD4D4477373
Gy＝6DCB7718C64642231DF10C935FD7E8BCBCF23C87CDC1BFD857DCDDEF
n＝FFFFFFFFFFFFFFFEFFFFFFFF0001096006D4A812714381686D1BCD29
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝EDE90BC2317ACEA4FAC05394C10FF1972748E806C8CB2F151893E97C
Gx＝3E115B26D2299312B5E149177CD87313BBEB68FC6529FE56DAF4C347
Gy＝76C24DB215839F4890FD3B5B6ED3B0FFD0443206771BE3359432AFA3
n＝FFFFFFFEFFFFFFFFFFFFFFFFFFFFC470EE78E46462D1559CBAB1684B
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝18DFDB909315F8050B291DBFD6D79EE347B125BF815C3D43A9D53FDA
Gx＝1B66919C660D91263C3EDA3BE43C40F8F7470CD386DA901A30B36EC8
Gy＝136F80185FCFF03C6D204BDC2AAA97B93BD7534DEDDEB56B0BE98F9D
n＝FFFFFFFEFFFFFFFFFFFFFFFFFFFE16E4C6603718C374BC2C403A3E3D
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝65D470FFA6063282AC80670FD970A057472F844D5288548203C10B14
Gx＝7597861368B6BD2D15BBB4D44B5EFF2F26EC5F24D495224E7F163FE
Gy＝247D42457323B85C22C94B9BF497739BF0135F66A9BB5BAA4AADFCB6
n＝FFFFFFFEFFFFFFFFFFFFFFFFFFFFB09B25199C3C45A15F7145FDAC0D
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝6239851169AF336253D6B269966CABF07DE36B941D4EDE1D578454F3
Gx＝7E8342386104A93D329064F9F168F65A8D4AC7438CA347A79C1C2748
Gy＝CA8FEC052F6C7101DD760C4DFF9F8DFBC6BB7CC420A3E06627D156B9
n＝FFFFFFFEFFFFFFFFFFFFFFFFFFFF855D60E9C373CC8FE23AA6FC336B
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝2700FF77B1235CAF22A35F20D78AB7B04D12FF4006DD9FC6525840B6
Gx＝7A4F3469E3E07967BA4C1729C86EB4AD3071612965CF88A094C9184C
Gy＝137E03C2068E68721CE71A000D88C0070B9BFD71EF18B027E4A79ACB
n＝FFFFFFFFFFFFFFFFFFFFFFFFFFFEE107B67D487E68FB6E672C2DB031
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝67E0F23C36BDA999E8D7AB3A9EF31BAB11B19C247901C7F402720015
Gx＝615FBFCC9C14291BEA3EA547DC3FE26DA0D2EB427A5436758E9686E2
Gy＝1B51AB3ED9536850335F268E240C8BFFF691C4421058C2358C9B75DF
n＝1000000000000000000000000000113AE63259EE823A1360A1934BCFF
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝9A88C3406278CBCBA1688FFA6C4371BE472213999038393A2369E1C2
Gx＝18A7FC17C22CB358FBDFE391CF0749DE8CDB4E6DE7B5E0BD75F874AA
Gy＝10AA2BF80C9993AD57AFA479CB3DADA382F76742554B4C8E471A1980
n＝FFFFFFFFFFFFFFFFFFFFFFFFFFFF32D37DA7EFB23D7840C0E7675041
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝8B22A7551C6ED668A52E767676F5FD390452940FA5F24221E9654686
Gx＝7BCD48550A57A5ABC01B99177E3E292B22B8051B5FCA9A302AE80D45
Gy＝919DF9F61BAC799180D38E5879BFDCC8ABB7179597DAE4B036585945
n＝100000000000000000000000000001C3620B759F9F761AAFD3DD33B89。
6.一种椭圆曲线加解密参数选择装置，该参数选择装置选择椭圆曲线加解密基域为素域Fp，其中p为224位，并且p满足 p = 2 224 + Σ i = 0 6 2 32 i k i , ki∈{-1，0，1}或p满足p＝2224-2k-1，1≤k≤223。
7.如权利要求6的椭圆曲线加解密参数选择装置，进一步选择在素域Fp上的椭圆曲线具有Weierstrass方程：y2＝x3+ax+b，a、b∈Fp，其中，选择a≡-3mod p，并且a和b满足(4a3+27b2)mod p≠0，同时a和b的选取使得椭圆曲线的阶n为素数。
8.如权利要求6的椭圆曲线加解密参数选择装置，进一步选择素数p为
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
p＝FFFFFFFFFFFFFFFEFFFFFFFF0000000100000000FFFFFFFEFFFFFFFF
p＝FFFFFFFF000000000000000000000000000000000000000100000001
p＝FFFFFFFF0000000000000001000000000000000000000000FFFFFFFF
p＝FFFFFFFF000000010000000100000001000000000000000000000001
p＝FFFFFFFF000000010000000100000000FFFFFFFEFFFFFFFF00000001
p＝FFFFFFFEFFFFFFFF000000000000000000000000FFFFFFFFFFFFFFFF
p＝FFFFFFFEFFFFFFFF00000000FFFFFFFEFFFFFFFF00000000FFFFFFFF
p＝FFFFFFFEFFFFFFFEFFFFFFFF00000000000000010000000100000001
p＝FFFFFFFEFFFFFFFEFFFFFFFEFFFFFFFF00000000FFFFFFFF00000001
p＝FFFFFFFEFFFFFFFEFFFFFFFEFFFFFFFEFFFFFFFF0000000100000001
p＝FFFFFFFFFFFFFF  FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF。
9.如权利要求7的椭圆曲线加解密参数选择装置，进一步选择椭圆曲线参数p、a和b为以下组中的任意一个：
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝DCB4307F23CA5F944E9F23CF61A0D1F6FD3708A87B7AE976716E68B
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝35F23F53A51F4D9BB689A7071A9218EA426948D22FC1CCBABF59F70
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝67DD503F0045DC305551E3C071C757D9D189352DB4FBE1C0F52444B2
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝B4C9085AE39EF3E1638A2540D0B367B7FBAC9FDF3908DD8AD0646FEB
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝1248940303D72DE541E65E152131EDB4DD0728CF10EA33F4F07AA8F9
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝EDE90BC2317ACEA4FAC05394C10FF1972748E806C8CB2F151893E97C
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝18DFDB909315F8050B291DBFD6D79EE347B125BF815C3D43A9D53FDA
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝65D470FFA6063282AC80670FD970A057472F844D5288548203C10B14
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝6239851169AF336253D6B269966CABF07DE36B941D4EDE1D578454F3
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝2700FF77B1235CAF22A35F20D78AB7B04D12FF4006DD9FC6525840B6
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝67E0F23C36BDA999E8D7AB3A9EF31BAB11B19C247901C7F402720015
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝9A88C3406278CBCBA1688FFA6C4371BE472213999038393A2369E1C2
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝8B22A7551C6ED668A52E767676F5FD390452940FA5F24221E9654686。
10.如权利要求9的椭圆曲线加解密参数选择装置，进一步选择各组中椭圆曲线的阶n和基点G＝(Gx，Gy)分别为：
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝DCB4307F23CA5F944E9F23CF61A0D1F6FD3708A87B7AE976716E68B
Gx＝1011D487D3D6F256A3FBF97F32CACE82DA9D26EE6BBCA871EE045DF8
Gy＝EAB80678DB3B1929B4A779CBBBD5543667C9FE9915847A8CC13093CF
n＝FFFFFFFFFFFFFFFEFFFFFFFEFFFF3DED44B6BC3AF1E7FED549DB388B
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝35F23F53A51F4D9BB689A7071A9218EA426948D22FC1CCBABF59F70
Gx＝1608C085BA6DC7ECEA756615C488ED797BC9C4FBC9118CAD8E4A6415
Gy＝5B0646D2BB213BE1FCFE9801CC89983FD89DD0D3E22F44F33CF688ED
n＝FFFFFFFFFFFFFFFEFFFFFFFEFFFF58602970D61D570A6FA14E32DDEB
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝67DD503F0045DC305551E3C071C757D9D189352DB4FBE1C0F52444B2
Gx＝2FDA3CFE677DB3F91B08C6D3B45652309C9B460DB629BE741F9F6B3
Gy＝1BD0D2FA2991917F809663F7D6A6557EDF933007E67615C35F8BDED3
n＝FFFFFFFFFFFFFFFEFFFFFFFEFFFE579C1262BBBD93EE592F2D864B45
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝B4C9085AE39EF3E1638A2540D0B367B7FBAC9FDF3908DD8AD0646FEB
Gx＝22648F44B91CB42E444FDF1B07AC0EF529FE45F9661EF4A1FABEC067
Gy＝2DE2F870C74D356FE415DB1920E601C164895C46FF677E26B09D3925
n＝FFFFFFFFFFFFFFFEFFFFFFFEFFFE2DEA1667F52001AC41EC4A4385F1
p＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFF
a＝FFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFFFFFFFFFEFFFFFFFEFFFFFFFC
b＝1248940303D72DE541E65E152131EDB4DD0728CF10EA33F4F07AA8F9
Gx＝4B2C9376020B9DE449768FA87E9FA0862DF0BBD34A36FDD4D4477373
Gy＝6DCB7718C64642231DF10C935FD7E8BCBCF23C87CDC1BFD857DCDDEF
n＝FFFFFFFFFFFFFFFEFFFFFFFF0001096006D4A812714381686D1BCD29
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝EDE90BC2317ACEA4FAC05394C10FF1972748E806C8CB2F151893E97C
Gx＝3E115B26D2299312B5E149177CD87313BBEB68FC6529FE56DAF4C347
Gy＝76C24DB215839F4890FD3B5B6ED3B0FFD0443206771BE3359432AFA3
n＝FFFFFFFEFFFFFFFFFFFFFFFFFFFFC470EE78E46462D1559CBAB1684B
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝18DFDB909315F8050B291DBFD6D79EE347B125BF815C3D43A9D53FDA
Gx＝1B66919C660D91263C3EDA3BE43C40F8F7470CD386DA901A30B36EC8
Gy＝136F80185FCFF03C6D204BDC2AAA97B93BD7534DEDDEB56B0BE98F9D
n＝FFFFFFFEFFFFFFFFFFFFFFFFFFFE16E4C6603718C374BC2C403A3E3D
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝65D470FFA6063282AC80670FD970A057472F844D5288548203C10B14
Gx＝7597861368B6BD2D15BBB4D44B5EFF2F26EC5F24D495224E7F163FE
Gy＝247D42457323B85C22C94B9BF497739BF0135F66A9BB5BAA4AADFCB6
n＝FFFFFFFEFFFFFFFFFFFFFFFFFFFFB09B25199C3C45A15F7145FDAC0D
p＝FFFFFFFF000000000000000000000000000000000000000100000001
a＝FFFFFFFF0000000000000000000000000000000000000000FFFFFFFE
b＝6239851169AF336253D6B269966CABF07DE36B941D4EDE1D578454F3
Gx＝7E8342386104A93D329064F9F168F65A8D4AC7438CA347A79C1C2748
Gy＝CA8FEC052F6C7101DD760C4DFF9F8DFBC6BB7CC420A3E06627D156B9
n＝FFFFFFFEFFFFFFFFFFFFFFFFFFFF855D60E9C373CC8FE23AA6FC336B
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝2700FF77B1235CAF22A35F20D78AB7B04D12FF4006DD9FC6525840B6
Gx＝7A4F3469E3E07967BA4C1729C86EB4AD3071612965CF88A094C9184C
Gy＝137E03C2068E68721CE71A000D88C0070B9BFD71EF18B027E4A79ACB
n＝FFFFFFFFFFFFFFFFFFFFFFFFFFFEE107B67D487E68FB6E672C2DB031
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝67E0F23C36BDA999E8D7AB3A9EF31BAB11B19C247901C7F402720015
Gx＝615FBFCC9C14291BEA3EA547DC3FE26DA0D2EB427A5436758E9686E2
Gy＝1B51AB3ED9536850335F268E240C8BFFF691C4421058C2358C9B75DF
n＝1000000000000000000000000000113AE63259EE823A1360A1934BCFF
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝9A88C3406278CBCBA1688FFA6C4371BE472213999038393A2369E1C2
Gx＝18A7FC17C22CB358FBDFE391CF0749DE8CDB4E6DE7B5E0BD75F874AA
Gy＝10AA2BF80C9993AD57AFA479CB3DADA382F76742554B4C8E471A1980
n＝FFFFFFFFFFFFFFFFFFFFFFFFFFFF32D37DA7EFB23D7840C0E7675041
p＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFF
a＝FFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF7FFFFFFFC
b＝8B22A7551C6ED668A52E767676F5FD390452940FA5F24221E9654686
Gx＝7BCD48550A57A5ABC01B99177E3E292B22B8051B5FCA9A302AE80D45
Gy＝919DF9F61BAC799180D38E5879BFDCC8ABB7179597DAE4B036585945
n＝100000000000000000000000000001C3620B759F9F761AAFD3DD33B89。
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* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101567783B (zh) * 2008-04-24 2012-08-22 深圳市同洲电子股份有限公司 一种基于ⅱ型高斯基域的椭圆曲线加解密方法和装置
CN107682151A (zh) * 2017-10-30 2018-02-09 武汉大学 一种gost数字签名生成方法及系统
CN107707358A (zh) * 2017-10-30 2018-02-16 武汉大学 一种ec‑kcdsa数字签名生成方法及系统

Cited By (5)

* Cited by examiner, † Cited by third party
Publication number Priority date Publication date Assignee Title
CN101567783B (zh) * 2008-04-24 2012-08-22 深圳市同洲电子股份有限公司 一种基于ⅱ型高斯基域的椭圆曲线加解密方法和装置
CN107682151A (zh) * 2017-10-30 2018-02-09 武汉大学 一种gost数字签名生成方法及系统
CN107707358A (zh) * 2017-10-30 2018-02-16 武汉大学 一种ec‑kcdsa数字签名生成方法及系统
CN107707358B (zh) * 2017-10-30 2019-12-24 武汉大学 一种ec-kcdsa数字签名生成方法及系统
CN107682151B (zh) * 2017-10-30 2021-02-02 武汉大学 一种gost数字签名生成方法及系统

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