Download PDF by Raphael Pass, Abhi Shelat: A Course in Cryptography

By Raphael Pass, Abhi Shelat

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Given such a pair p, (q1 , . . , qk ), one can use a central result from group theory to test whether an element is a generator. For example, in the case of safe primes, testing whether an element g ∈ Z∗p is a generator consists of checking g = ±1 mod p and gq = 1 mod p. As we will see later, the collection DL is also a special collection of one-way functions in which each function is a permutation. 9 RSA Collection Another popular assumption is the RSA Assumption. The RSA Assumption implies the Factoring assumption; it is not known whether the converse is true.

Ym ). If f (zi ) = y, then output zi ; otherwise, fail and output ⊥. To improve our chances of inverting f , we will run A0 several times using independently chosen random coins. Define the algorithm A : {0, 1}n → {0, 1}n ∪ ⊥ to run A0 with its input 2nm2 p(n) times and output the first non-⊥ result it receives. If all runs of A0 result in ⊥, then A also outputs ⊥. ” Note that the probability that A fails to invert f ( x ) on a good x is small: Pr [A( f ( x )) fails | x ∈ Gn ] ≤ 1 1− 2 2m p(n) 2m2 np(n) ≈ e−n .

Unfortunately, the second phenomena is on rare occasion false. Despite there rarity (starting with 561, 1105, 1729, . , there are only 255 such cases less than 108 ), there are an infinite number of counter examples collectively known as the Carmichael numbers. Thus, for correctness, our procedure must handle these rare cases. To do so, we add a second check that verifies that none of the intermediate powers of a encountered during the modular exponentiation computation of an−1 are non-trivial square-roots of 1.

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A Course in Cryptography by Raphael Pass, Abhi Shelat

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