This part of GM/T 32918 specifies the necessary mathematical basics and related cryptographic techniques involved in public key cryptographic algorithm SM2 based on elliptic curves to help implement the cryptographic mechanisms specified in other parts.

This part applies to the elliptic curve public key cryptography algorithm with the base field being the prime field and the binary extension field.

2 Symbols and abbreviations

For purpose this part, the following symbols and abbreviations apply.

B: MOV threshold. Positive B, so that the obtaining the discrete logarithm of the number on is at least as difficult as obtaining the discrete logarithm of elliptic curve on Fq.

deg (f): the power of the polynomial f(x).

E: An elliptic curve defined by a and b on the finite field.

E (Fq): a set of all rational points of the elliptic curve E on Fq (including the infinity point O).

ECDLP: Elliptic curve discrete logarithm problem.

Fp: a prime field containing p elements.

Fq: a finite field containing q elements.

: a multiplicative group consisting of all non-zero elements in Fq.

: a binary extension containing 2 m elements.

G: A base point of an elliptic curve whose order is a prime.

gcd (x,y): the greatest common factor of x and y.

h: cofactor, h - E(Fq)/n, where n is the order of the base point G.

LeftRotate (): Looped left shift operation.

lmax: the upper bound of the largest prime factor of the cofactor h.

m: the number of extensions of binary extension regarding F2 .

mod f(x): the operation of modulus polynomial f(x). If f(x) is a polynomial on a binary field, then all coefficients perform modulo-2 operation.

mod n: modulo-n operation. For example, 23 mod 7-2:

n: the order of the base point G (n is the prime element of E (Fq)).

GB/T 32918.1-2016 The following standards are cited：

GB/T 32918.1-2016 Cited by the following standards：