This chapter introduces the basic properties of congruences modulo n, along with the related notion of congruence classes modulo n. Other items discussed include the Chinese remainder theorem, Euler's phi function, arithmetic functions and Möbius inversion, and Fermat's little theorem.

Definitions and basic properties

For positive integer n, and for a, b ∈ ℤ, we say that a is congruent tobmodulon if n | (a - b), and we write a ≡ b (mod n). If n ∣ (a - b), then we write a ≢ b (mod n). The relation a ≡ b (mod n) is called a congruence relation, or simply, a congruence. The number n appearing in such congruences is called the modulus of the congruence. This usage of the “mod” notation as part of a congruence is not to be confused with the “mod” operation introduced in §1.1.

A simple observation is that a ≡ b (mod n) if and only if there exists an integer c such that a = b + cn. From this, and Theorem 1.4, the following is immediate:

Theorem 2.1.Let n be a positive integer. For every integer a, there exists a unique integer b such that a ≡ b (mod n) and 0 ≤ b < n, namely, b ≔ a mod n.

This chapter develops a number of other concepts concerning rings. These concepts will play important roles later in the text, and we prefer to discuss them now, so as to avoid too many interruptions of the flow of subsequent discussions.

Algebras

Let R be a ring. An R-algebra (or algebra overR) is a ring E, together with a ring homomorphism τ: R → E. Usually, the map τ will be clear from context, as in the following examples.

Example 17.1. If E is a ring that contains R as a subring, then E is an R-algebra, where the associated map τ: R → E is just the inclusion map.

Example 17.2. Let E1, …, En be R-algebras, with associated maps τi: R → Ei, for i = 1, …, n. Then the direct product ring E:= E1 × … × En is naturally viewed as an R-algebra, via the map τ that sends a ∈ R, to (τ1(a), …, τn(a)) ∈ E.

Example 17.3. Let E be an R-algebra, with associated map τ: R → E, and let I be an ideal of E. Consider the quotient ring E/I. If ρ is the natural map from E onto E/I, then the homomorphism ρ ∘ τ makes E/I into an R-algebra, called the quotient algebra ofEmoduloI.

As we have seen in Theorem 9.16, for a prime is a cyclic group of order p - 1. This means that there exists a generator, such that for all, α can be written uniquely as α = γx, where x is an integer with 0 ≤ x < p - 1; the integer x is called the discrete logarithm of α to the base γ, and is denoted logγ α.

This chapter discusses some computational problems in this setting; namely, how to efficiently find a generator γ, and given γ and α, how to compute logγ α.

More generally, if γ generates a subgroup G of of order q, where q | (p - 1), and α ∈ G, then logγ α is defined to be the unique integer x with 0 ≤ x < q and α = γx. In some situations it is more convenient to view logγ α as an element of ℤq. Also for x ∈ ℤq, with x = [a]q, one may write γx to denote γa. There can be no confusion, since if x = [a′]q, then γa′ = γa. However, in this chapter, we shall view logγ α as an integer.

Although we work in the group, all of the algorithms discussed in this chapter trivially generalize to any finite cyclic group that has a suitably compact representation of group elements and an efficient algorithm for performing the group operation on these representations.

We establish here a few notational conventions used throughout the text.

Arithmetic with ∞

We shall sometimes use the symbols “∞” and “–∞” in simple arithmetic expressions involving real numbers. The interpretation given to such expressions is the usual, natural one; for example, for all real numbers x, we have -∞ < x < ∞, x + ∞ = ∞, x - ∞ = -∞, ∞ + ∞ = ∞, and (-∞) + (-∞) = -∞. Some such expressions have no sensible interpretation (e.g., ∞-∞).

Logarithms and exponentials

We denote by log x the natural logarithm of x. The logarithm of x to the base b is denoted logbx.

We denote by ex the usual exponential function, where e ≈ 2.71828 is the base of the natural logarithm. We may also write exp[x] instead of ex.

Sets and relations

We use the symbol ∅ to denote the empty set. For two sets A, B, we use the notation A ⊆ B to mean that A is a subset of B (with A possibly equal to B), and the notation A ⊆ B to mean that A is a proper subset of B (i.e., A ⊆ B but A ≠ B); further, A ∪ B denotes the union of A and B, A ∩ B the intersection of A and B, and A \ B the set of all elements of A that are not in B.