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One-way functions and circuit complexity (1987)
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| Venue: | Information and Computation |
| Citations: | 13 - 0 self |
BibTeX
@ARTICLE{Lagarias87one-wayfunctions,
author = {J. C. Lagarias and J. C. Lagarias},
title = {One-way functions and circuit complexity},
journal = {Information and Computation},
year = {1987},
pages = {226--240}
}
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Abstract
A finite function f is a mapping of {1, 2,..., m} into {1, 2,..., m} ∪ {#} where # is a symbol to be thought of as ‘‘undefined.’ ’ This paper defines a measure M ( f) of the difficulty of inverting a finite function f, which is given by M ( f) = MIN ⎧ log 2 C(g) _ ________ _ : g an inverse of f⎬ ⎭ where C ( f) is a circuit log 2 C ( f) complexity measure of the difficulty of computing f. We say that one-way functions exist (in a circuit complexity sense) if and only if M ( f) is unbounded. We prove that one-way functions exist if and only if the satisfiability problem SAT has polynomial sized circuits. This paper also defines an analogous measure M d ( f) in which only circuits of depth ≤ d are allowed. We show that one-way functions exist in this bounded-depth circuit complexity model, by showing for the permutations σ n on {1, 2,..., 2 n} defined by σ n (k) ≡ 3k (mod 2 n) that for d ≥ 4 there is a positive constant c d such that M d (σ n)> c d log n as n → ∞.ONE-WAY FUNCTIONS AND CIRCUIT COMPLEXITY
