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By Raphael Pass, Abhi Shelat

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2. This definition is also closed under composition which may simplify reasoning in certain proofs. 3. e. ” 4. Our experience indicates that “natural” functions that are not known to be computable in polynomial-time require much more time to compute, so the separation we propose seems well-founded. Note, our treatment of computation is an asymptotic one. In practice, concrete running time needs to be considered carefully, as do other hidden factors such as the size of the description of A. Thus, when porting theory to practice, one needs to set parameters carefully.

2 Randomized Computation A natural extension of deterministic computation is to allow an algorithm to have access to a source of random coin tosses. Allowing this extra freedom is certainly plausible (as it is conceivable to generate such random coins in practice), and it is believed to enable more efficient algorithms for computing certain tasks. Moreover, it will be necessary for the security of the schemes that we present later. For example, as we discussed in chapter one, Kerckhoff’s principle states that all algorithms in a scheme should be public.

Easy to compute. t. C that computes f ( x ) on all inputs x ∈ {0, 1}∗ , and 2. ). , those where inverting the function for most x values is easy, but every machine fails to invert f ( x ) for infinitely many x’s. It is an open question whether such functions can still be used for good encryption schemes. This observation motivates us to refine our requirements. We want functions where for a randomly chosen x, the probability that we are able to invert the function is very small. With this new definition in mind, we begin by formalizing the notion of very small.

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A Course in Cryptography by Raphael Pass, Abhi Shelat

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