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

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By Raphael Pass, Abhi Shelat

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This notion is central to modern definitions of security for encryption schemes, but also for formally defining notions such as pseudo-random generation, commitment schemes, zero-knowledge protocols, etc. Knowledge. A central desideratum in the design of cryptographic protocols is to ensure that the protocol execution does not leak more “knowledge” than what is necessary. In this part of the course, we investigate “knowledge-based” (or rather zero knowledge-based) definitions of security. Authentication.

A runs in polynomial time (or is an efficient randomized algorithm) if there exists a constant c such that A runs in time T (n) = nc . Finally, we must also extend our notion of computation to randomized algorithms. In particular, once an algorithm has a random tape, its output becomes a distribution over some set. In the case of deterministic computation, the output is a singleton set, and this is what we require here as well. 6 A randomized algorithm A computes a function f : {0, 1}∗ → {0, 1}∗ if for all x ∈ {0, 1}∗ , A on input x, outputs f ( x ) with probability 1.

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 .

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