ISO/IEC 18370-2:2016 情報技術—セキュリティ技術—ブラインドデジタル署名—パート2：離散対数ベースのメカニズム | ページ 6

3.1

abelian group

group (G, *) such that a * b = b * a for every a and b in G

3.2

cyclic group

group G of n elements that contains an element a in G, called the generator, of order n

[SOURCE: ISO/IEC 14888‑3:2006, 3.2]

3.3

elliptic curve over a finite field

set E of points P = (x, y), where x and y are elements of the finite field (3.6) , that satisfy a certain equation, together with an extra point referred to as the point at infinity

Note 1 to entry: In this part of ISO/IEC 18370, only finite fields containing exactly q elements for a prime q > 3 are considered. In this case, the equation that every point P = (x, y) of E (other than the point at infinity) should satisfy is of the form y² = x³ + ax + b. The finite field elements a and b should satisfy 4a³ + 27b² ≠ 0 _F (where 0 _F is the additive identity element of the finite field).

Note 2 to entry: The set of points E, together with an appropriately defined operation, forms a finite commutative group (3.5) , where the point at infinity is the identity element.

3.4

field

set of elements S and a pair of operations (+,*) defined on S, such that: i) a * (b + c) = a * b + a * c for every a, b and c in S, ii) S together with + forms an abelian group (3.1) (with identity element 0), and iii) S excluding 0 together with * forms an abelian group

3.5

finite commutative group

abelian group (3.1) (G, *) with a finite number of elements

Note 1 to entry: If a⁰ = e, and aⁿ⁺¹ = a * aⁿ (for n ≥ 0) is defined recursively, the order of a ∈ G is the least positive integer n, such that aⁿ = e.

Note 2 to entry: In some cases, such as when G is the set of points on an elliptic curve, arithmetic in the finite set G is described using additive notation.

3.6

finite field

field (3.4) such that the underlying set of elements is finite

Note 1 to entry: For any positive integer, m and a prime p, there exists a finite field containing exactly q = p^m elements. This field is unique up to an isomorphism and is denoted by F_q .

[SOURCE: ISO/IEC 18033‑2:2006, 3.21]

3.7

group

set of elements G and an operation * defined on the set of elements such that: i) (a * b) * c = a * (b * c) for every a, b and c in G, ii) there exists an identity element, e in G, such that a * e = e * a = a for every a in G, and iii) for every a in G, there exists an inverse element, a^-1 in G, such that a * a^-1 = a^-1 * a = e

3.8

security parameters

variables that determine the security strength of a mechanism