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GM/T 0003.1-2012: Public key cryptographic algorithm SM2 based on elliptic curves - Part 1: General
GM/T 0003.1-2012
Public key cryptographic algorithm SM2 based on elliptic curves-Part 1.General
ICS 35.040
L80
Record number. 36826-2012
People's Republic of China Password Industry Standard
SM2 elliptic curve public key cryptographic algorithm
Part 1.General
Part 1.General
Released on.2012-03-21
2012-03-21 Implementation
Issued by the National Cryptography Administration
Table of contents
Foreword Ⅰ
Introduction Ⅱ
1 Scope 1
2 Symbols and abbreviations 1
3 Domain and elliptic curve 2
3.1 Finite Field 2
3.2 Elliptic curve on a finite field 3
4 Data types and their conversion 5
4.1 Data Type 5
4.2 Data type conversion 5
5 Elliptic curve system parameters and their verification 8
5.1 General requirements 8
5.2 Parameters and verification of elliptic curve system on Fp 8
5.3 Elliptic curve system parameters on F2m and their verification 9
6 Key pair generation and public key verification 9
6.1 Key pair generation 9
6.2 Verification of the public key 9
Appendix A (informative appendix) Background knowledge about elliptic curves 11
A.1 Prime domain Fp 11
A.2 Binary extension F2m 13
A.3 Calculation of multiple points on elliptic curve 23
A.4 Method for solving the discrete logarithm problem of elliptic curve 26
A.5 Compression of points on elliptic curves 27
Appendix B (informative appendix) Number theory algorithm 29
B.1 Finite fields and modular operations 29
B.2 Polynomials over finite fields 33
B.3 Elliptic Curve Algorithm 35
Appendix C (informative appendix) Curve example 37
C.1 General requirements 37
C.2 Elliptic curve on Fp 37
C.3 F2m upper elliptic curve 37
Appendix D (informative appendix) Quasi-random generation and verification of elliptic curve equation parameters 39
D.1 Quasi-random generation of elliptic curve equation parameters 39
D.2 Verification of the parameters of the elliptic curve equation 40
Reference 41
Preface
GM/T 0003-2012 "SM2 Elliptic Curve Public Key Cryptography Algorithm" is divided into 5 parts.
---Part 1.General Provisions;
---Part 2.Digital Signature Algorithm;
---Part 3.Key Exchange Protocol;
---Part 4.Public key encryption algorithm;
---Part 5.Parameter definition.
This part is part 1 of GM/T 0003.
This part is drafted in accordance with the rules given in GB/T 1.1-2009.
Please note that certain contents of this document may involve patents. The issuing agency of this document is not responsible for identifying these patents.
Appendix A, Appendix B, Appendix C and Appendix D of this section are informative appendices.
This part is proposed and managed by the State Cryptography Administration.
Drafting organizations of this section. Beijing Huada Xin'an Technology Co., Ltd., Chinese People's Liberation Army Information Engineering University, Chinese Academy of Sciences
Communication Protection Research and Education Center.
The main drafters of this section. Chen Jianhua, Zhu Yuefei, Ye Dingfeng, Hu Lei, Pei Dingyi, Peng Guohua, Zhang Yajuan, Zhang Zhenfeng.
introduction
N.Koblitz and V.Miler independently proposed the application of elliptic curves to public key cryptosystems in 1985.Elliptic curve
The properties of the curve based on the key cryptography are as follows.
---The elliptic curve on the finite field forms a finite commutative group under the point addition operation, and its order is similar to the scale of the base field;
---Similar to the power operation in the finite field multiplication group, the elliptic curve multiple point operation constitutes a one-way function.
In the calculation of multiple points, the problem of knowing the multiple point and the base point to solve the multiple is called the elliptic curve discrete logarithm problem. For general ellipse
For the discrete logarithm of the curve, there are only exponential computational complexity solutions at present. And the decomposition of large numbers and discrete pairs over finite fields
Compared with the number problem, the elliptic curve discrete logarithm problem is much more difficult to solve. Therefore, under the same safety requirements, the elliptic curve is dense
The key size required for the code is much smaller than other public key ciphers.
This section describes the basic mathematical knowledge and general techniques necessary to help implement the cryptographic mechanism specified in the other sections.
SM2 elliptic curve public key cryptographic algorithm
Part 1.General
1 Scope
This part of GM/T 0003 gives the necessary basic mathematical knowledge and related cryptographic techniques involved in the SM2 elliptic curve public key cryptographic algorithm.
In order to help realize the cryptographic mechanism specified in other parts.
This part is applicable to elliptic curve public key cryptographic algorithms whose base domain is prime domain and binary extension domain.
2 Symbols and abbreviations
The following symbols and abbreviations apply to this section.
a, b. elements in Fq, they define an elliptic curve E on Fq.
B. MOV threshold. The positive number B makes finding the discrete logarithm of FqB at least as difficult as finding the discrete logarithm of the elliptic curve on Fq.
deg(f). The degree of polynomial f(x).
E. An elliptic curve defined by a and b on a finite field.
E(Fq). The set of all rational points (including the infinity point O) of the elliptic curve E on Fq.
ECDLP. Discrete logarithm problem of elliptic curve.
Fp. A prime domain containing p elements.
Fq. A finite field containing q elements.
F*q. The multiplicative group formed by all non-zero elements in Fq.
F2m. A binary extension field containing 2m elements.
G. A base point of the elliptic curve, whose order is a prime number.
gcd(x,y). The greatest common factor of x and y.
h. cofactor, h=
O. A special point on the elliptic curve, called the infinity point or zero point, is the identity element of the additive group of the elliptic curve.
P.P=(xP,yP) is a point on the elliptic curve except O, and its coordinates xP,yP satisfy the elliptic curve equation.
P1 P2.The sum of two points P1 and P2 on the elliptic curve E.
p. A prime number greater than 3.
q. The number of elements in the finite field Fq.
rmin. the lower bound of order n of the base point G.
Tr(). Trace function.
xP. The x coordinate of point P.
x-1modn. The unique integer y that makes x·y≡1(modn) true, 1≤y≤n-1, gcd(x,n)=1.