By VICTOR SHOUP
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Extra resources for A COMPUTATIONAL INTRODUCTION TO NUMBER THEORY AND ALGEBRA (VERSION 1)
5 holds if and only if z ≡ z (mod n). ✷ Suppose that a, b, n ∈ Z with n > 0, a = 0, and gcd(a, n) = 1. This theorem says that there exists a unique integer z satisfying az ≡ b (mod n) and 0 ≤ z < n. Setting s := b/a ∈ Q, we may generalize the “mod” operation, deﬁning s mod n to be this value z. As the reader may easily verify, this deﬁnition of s mod n does not depend on the particular choice of fraction used to represent the rational number s. With this notation, we can simply write a−1 mod n to denote the unique multiplicative inverse of a modulo n that lies in the interval 0, .
Theoretical computer scientists sometimes equate the notion of an “eﬃcient” algorithm with that of a polynomial-time algorithm (although not everyone takes theoretical computer scientists very seriously, especially on this point). A polynomial-time algorithm is one whose running time on inputs of length n is bounded by nc + d for some constants c and d (a “real” theoretical computer scientist will write this as nO(1) ). To make this notion mathematically precise, one needs to deﬁne the length of an algorithm’s input.
Next, we deﬁne a binary operation on arithmetic functions that has a number of interesting properties and applications. Let f and g be arithmetic functions. The Dirichlet product of f and g, denoted f g, is the arithmetic function whose value at n is deﬁned by the formula (f f (d)g(n/d), g)(n) := d|n the sum being over all positive divisors d of n. Another, more symmetric, way to write this is (f g)(n) = f (d1 )g(d2 ), n=d1 d2 the sum being over all pairs (d1 , d2 ) of positive integers with d1 d2 = n.
A COMPUTATIONAL INTRODUCTION TO NUMBER THEORY AND ALGEBRA (VERSION 1) by VICTOR SHOUP