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By Firoz Kaderali

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Extra resources for Foundations and Applications of Cryptology [Lecture notes]

Example text

Hence, we conclude that the congruence modulo n is an equivalence relation on the set of all integers. We now give some useful properties of congruences modulo n which are easy to prove. For all a, b, c, d ∈ Z, if a ≡ b mod n and c ≡ d mod n, then the following holds: 1. −a ≡ −b mod n. 2. a + c ≡ b + d mod n. 3. a · c ≡ b · d mod n. 4. a/c ≡ b/d mod n. In item 3 of the above listed properties, if we consider the special case (a = c and b = d), we obtain a2 ≡ b2 mod n. Using the item 3 again with this special case (a = c and b = d) we get a3 ≡ b3 mod n.

An element a ∈ R of the ring (R, ∗, ◦) is invertible or unit, if it is invertible in (R, ◦). The set of unit elements in a ring (R, ∗, ◦) forms a group under the operation o which is called the group of the units of R and is denoted by (R∗ , ◦). 2-4: The set of integers Z with the usual operations of addition and multiplication is a commutative ring, since 1. (Z, +) is an abelian group with zero element 0, and 2. the group operation · is associative, that is, (a · b) · c = a · (b · c) for all a, b, c ∈ Z, and 3.

The order of the field Kq is the number q of its elements. Now, we will describe the structure of finite fields. First we introduce the notion of an irreducible polynomial, since irreducible polynomials are indispensible for constructing finite fields. Moreover, each polynomial in Kq [x] can be written as product of irreducible polynomials in an essentially unique manner. 4-3: Let f ∈ Kq [x] with deg(f ) ≥ 1. f is irreducible in Kq [x] (or prime in Kq [x]), if it cannot be expressed as a product of two polynomials in Kq [x], each of positive degree.

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