By Raphael Pass, Abhi Shelat

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Therefore, if there is some way to attack f , then that same method can be used (via the reduction) to break the original primitive f that we assume to be secure. The complete proof of Thm. 3 below. To introduce that proof, we first present two smaller examples. First, to gain familiarity with security reductions, we show a simple example of how to argue that if f is a strong one-way function, then g( x, y) = ( f ( x ), f (y)) is also a strong one-way function. 1. 2 If f is a strong one-way function, then g( x, y) = ( f ( x ), f (y)) is a strong one-way function.

1 (Algorithm) An algorithm is a deterministic Turing machine whose input and output are strings over alphabet Σ = {0, 1}. 2 (Running-time) An algorithm A is said to run in time T (n) if for all x ∈ {0, 1}∗ , A( x ) halts within T (| x |) steps. A runs in polynomial time if there exists a constant c such that A runs in time T (n) = nc . 3 (Deterministic Computation) An algorithm A is said to compute a function f : {0, 1}∗ → {0, 1}∗ if for all x ∈ {0, 1}∗ , A, on input x, outputs f ( x ). We say that an algorithm is efficient if it runs in polynomial time.

4 (Randomized (PPT) Algorithm) A randomized algorithm, also called a probabilistic polynomial-time Turing machine and abbreviated as PPT, is a Turing machine equipped with an extra random tape. Each bit of the random tape is uniformly and independently chosen. 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.

### A Course in Cryptography by Raphael Pass, Abhi Shelat

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