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notes-6 - Leo Reyzin Notes for BU CAS CS 538 1 6 General...

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Leo Reyzin. Notes for BU CAS CS 538. 1 6 General One-Way and Trapdoor Functions In this section, we will try to generalize what we’ve seen so far. For example, we know how to build a secure encryption out of RSA, but what exactly is RSA itself? In modern terms, it is a trapdoor permutation family, which we define below. 6.1 One-Way Functions Let us first introduce one-way functions. We’ve actually seen concrete examples of them before; this is just a generalization, so we can talk of a one-way function f independent of its particular implementation. Definition 1. A function f : { 0 , 1 } * { 0 , 1 } * is one-way if 1. it is polynomial-time computable; 2. it is hard to invert, i.e., for all probabilistic polynomial-time A there exists a negligible function η such that, for all k , Pr[ f ( A ( f ( x ) , 1 k )) = f ( x )] η ( k ), where the probability is taken over a random choice of k -bit string x and coin tosses of A . Note that it’s important that we are not requiring A to find x ; rather, any inverse of f ( x ) is fine. Of course, if f is a permutation (i.e., a bijective function), then it would be equivalent to require A to find x , because x is the only inverse of f ( x ). Note also the importance of selecting the input to A : the input is not selected uniformly at random; rather, x is selected uniformly at random, and the input is f ( x ). Of course, again, if f is a permutation, then the two are equivalent. An example is the following f : split the k -bit input into strings a of length k/ 2 and b of length k/ 2 , and output c = ab . The inverter A would have to find two large factors of c , which is believed to be hard. Note that the input c of A is not a uniformly selected integer; in particular, we know that it has two factors of (nearly) the same length.
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