GB/T 38635.1-2020Information security technology - Identity-based cryptographic algorithms SM9 - Part 1: General (English PDF)

1 Scope

This part of GB/T 38635 specifies the necessary relevant mathematical basics, cryptographic techniques and specific parameters involved in the identity-based cryptographic algorithms SM9.

This part is applicable to the implementation and application of identity-based cryptography SM9.

2 Normative references

The following referenced documents are indispensable for the application of this document. For dated references, only the edition cited applies. For undated references, the latest edition of the referenced document (including any amendments) applies.

GB/T 32905 Information security techniques - SM3 cryptographic hash algorithm

GB/T 32907 Information security technology - SM4 block cipher algorithm

3 Terms and definitions

For the purposes of this document, the following terms and definitions apply.

4 Symbols

For the purposes of this document, the following symbols apply.

cf: The cofactor of the elliptic curve order relative to N.

cid: A one-byte curve identifier that distinguishes the type of curve used.

deg(f): The degree of the polynomial f(x).

d1, d2: Two factors of k.

E: An elliptic curve defined over a finite field.

ECDLP: Elliptic curve discrete logarithm problem.

E(Fq): A set of all rational points (including the point at infinity, O) of the elliptic curve E over the finite field Fq.

E(Fq)[r]: A set of r-torsion points over E(Fq) [i.e., r-order torsion subgroup on the curve E(Fq)].

e: Bilinear pairing from G1×G2 to GT.

eid: A one-byte identifier of bilinear pairing e, which distinguishes the type of the bilinear pairing used.

FDLP: Finite field discrete logarithm problem.

Fp: A prime field containing p elements.

Fq: A finite field containing q elements.

F_q^*: A multiplicative group consisting of all non-zero elements in Fq.

F_(q^m ): m-degree extension field of finite field Fq.

GT: Multiplicative cyclic group with prime order N.

G1: Additive cyclic group with prime order N.

G2: Additive cyclic group with prime order N.

gcd(x, y): The greatest common divisor of x and y.

k: The embedding degree of the curve E(Fq) relative to N, where N is the prime factor of #E(Fq).

m: The degree of extension of the finite field F_(q^m ) relative to Fq.

modf(x): Operation of the modular polynomial f(x).

modn: Modulo n operation.

Example: 23 mod 7=2.

N: The order of cyclic groups G1, G2 and GT, which is a prime number greater than 2191.

O: A special point on an elliptic curve, called the point at infinity or zero point, which is the identity element in additive group of the elliptic curve.

P: P=(xP, yP) is a point on an elliptic curve other than O, of which the coordinates xP, yP satisfy the elliptic curve equation.

P1: The generating element of G1.

P2: The generating element of G2.

P+Q: The sum of two points P and Q on the elliptic curve E.

p: A prime number greater than 2191.

q: The number of elements in the finite field Fq.

xP: The x coordinate of point P.

x||y: The concatenation of x and y, where x and y are bit strings or byte strings.

x==y(mod q): x is congruent to y modulo q. That is, x mod q=y mod q.

yP: The y coordinate of point P.

#E(K): The number of points on E(K), called the order of the elliptic curve group E(K), where K is a finite field (including Fq and F_(q^k )).

: The cyclic group generated by point P on an elliptic curve.

[u]P: u-multiplied point of the point P on the elliptic curve.

[x, y]: A set of integers not less than x and not greater than y.