By Raphael Pass, Abhi Shelat

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Equivalently, a randomized algorithm is a Turing Machine that has access to a coin-tossing oracle that outputs a truly random bit on demand. To define efficiency we must clarify the concept of running time for a randomized algorithm. A subtlety arises because the run time of a randomized algorithm may depend on the particular random tape chosen for an execution. We take a conservative approach and define the running time as the upper bound over all possible random sequences. 5 (Running time) A randomized Turing machine A runs in time T (n) if for all x ∈ {0, 1}∗ , and for every random tape, 24 chapter 2.

We allow the adversary to be non-uniform. As before, we still allow the adversary to use random coins and require that the adversary’s running time is bounded by a polynomial. The primary motivation for using non-uniformity to model the adversary is to simplify definitions and proofs. u. ) A is a sequence of probabilistic machines A = { A1 , A2 , . } for which there exists a polynomial d such that the description size of | Ai | < d(i ) and the running time of Ai is also less than d(i ). We write A( x ) to denote the distribution obtained by running A| x| ( x ).

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.

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