By Raphael Pass, Abhi Shelat

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8 Discrete Logarithm-based Collection Another often used collection is based on the discrete logarithm problem in the group Z∗p for a prime p. 1 Discrete logarithm modulo p An instance ( p, g, y) of the discrete logarithm problem consists of a prime p and two elements g, y ∈ Z∗p . The task is to find an x such that g x = y mod p. , g = 1 or when p − 1 has many small prime factors), it is easy to either declare that no solution exists or solve the problem. However, when g is a generator of Z∗p , the problem is believed to be hard.

The trick is to define g(r1 , r2 ) to be i, f i ( x ) where i is generated using r1 as the random bits and x is sampled from Di using r2 as the random bits. The fact that g is a strong one-way function is left as an excercise. 6 Basic Computational Number Theory Before we can study candidate collections of one-way functions, it serves us to review some basic algorithms and concepts in number theory and group theory. 1 For N > 0 and a, b ∈ Z, 1. ( a mod N ) + (b mod N ) mod N ≡ ( a + b) mod N 2.

1. 2 If f is a strong one-way function, then g( x, y) = ( f ( x ), f (y)) is a strong one-way function. Proof. Suppose for the sake of reaching contradiction that g is not a strong one-way function. t. A and a polynomial p such that for infinitely many n, Pr ( x, y) ← {0, 1}2n ; z ← g( x, y) : A (12n , z) ∈ g−1 (z) ≥ 1 p(2n) 36 chapter 2. t. A that uses A in order to invert f on input u. In order to do this, A will choose a random y, compute v ← f (y) and then submit the pair (u, v) to A . Notice that this pair (u, v) is identically distributed as the pair 1 ( x, y) in the equation above.