By Raphael Pass, Abhi Shelat

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Despite there rarity (starting with 561, 1105, 1729, . , there are only 255 such cases less than 108 ), there are an infinite number of counter examples collectively known as the Carmichael numbers. Thus, for correctness, our procedure must handle these rare cases. To do so, we add a second check that verifies that none of the intermediate powers of a encountered during the modular exponentiation computation of an−1 are non-trivial square-roots of 1. This suggest the following approach known as the Miller-Rabin primality test presented by Miller [mil76] and Rabin [rab80].

Unfortunately, the second phenomena is on rare occasion false. Despite there rarity (starting with 561, 1105, 1729, . , there are only 255 such cases less than 108 ), there are an infinite number of counter examples collectively known as the Carmichael numbers. Thus, for correctness, our procedure must handle these rare cases. To do so, we add a second check that verifies that none of the intermediate powers of a encountered during the modular exponentiation computation of an−1 are non-trivial square-roots of 1.

1. Easy to compute. ) 2. Hard to invert. 3. Multiplication, Primes, and Factoring 29 Our eventual goal is to show that weak one-way functions can be used to construct strong one-way functions. Before showing this, let us consider some examples. 3 Multiplication, Primes, and Factoring In this section, we consider examples of one-way functions. A first candidate is the function f mult : N2 → N defined by f mult ( x, y) = 1 if x = 1 ∨ y = 1 x · y otherwise Is this a one-way function? Clearly, by the multiplication algorithm, f mult is easy to compute.