By Raphael Pass, Abhi Shelat

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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. 1 Some computationally “hard” problems Many commonly encountered functions are computable by efficient algorithms. However, there are also functions which are known or believed to be hard. Halting: The famous Halting problem is an example of an uncomputable problem: Given a description of a Turing machine M, determine whether or not M halts when run on the empty input.

T. Suppose we now feed A the product of a pair of random n-bit primes, z. e. the product of a pair of random n-bit numbers), A samples a pair ( x, y) uniformly, and replaces the product xy with the input z if both x and y are prime. 3), A fails to pass z to A with probability at most 1 − 4n1 2 . From Eq. 2), A fails to factor its input with probability at most 1/8n2 . Using the union bound, we conclude that A fails with probability at most 1− 1 4n2 + 1 1 ≤ 1− 2 8n2 8n for large n. In other words, A factors z with probability at least 1 for infinitely many n.

Thus, all that remains is to present a method to efficiently sample two random prime numbers. 17 above. Thus all four conditions in the definition of a one-way collection are satisfied. 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.

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A Course in Cryptography by Raphael Pass, Abhi Shelat

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