One-way permutations, computational asymmetry and distortion

Computational asymmetry, i.e., the discrepancy between the complexity of transformations and the complexity of their inverses, is at the core of one-way transformations. We introduce a computational asymmetry function that measures the amount of one-wayness of permutations. We also introduce the word-length asymmetry function for groups, which is an algebraic analogue of computational asymmetry. We relate boolean circuits to words in a Thompson monoid, over a fixed generating set, in such a way that circuit size is equal to word-length. Moreover, boolean circuits have a representation in terms of elements of a Thompson group, in such a way that circuit size is polynomially equivalent to word-length. We show that circuits built with gates that are not constrained to have fixed-length inputs and outputs, are at most quadratically more compact than circuits built from traditional gates (with fixed-length inputs and outputs). Finally, we show that the computational asymmetry function is closely related to certain distortion functions: The computational asymmetry function is polynomially equivalent to the distortion of the path length in Schreier graphs of certain Thompson groups, compared to the path length in Cayley graphs of certain Thompson monoids. We also show that the results of Razborov and others on monotone circuit complexity lead to exponential lower bounds on certain distortions.Comment: 33 page

One-Way Functions and Balanced NP

The existence of cryptographically secure one-way functions is related to the measure of a subclass of NP. This subclass, called BNP (``balanced NP\u27\u27), contains 3SAT and other standard NP problems. The hypothesis that BNP is not a subset of P is equivalent to the P \u3c\u3e NP conjecture. A stronger hypothesis, that BNP is not a measure 0 subset of E_2 = DTIME(2^polynomial) is shown to have the following two consequences. 1. For every k, there is a polynomial time computable, honest function f that is (2^{n^k})/n^k-one-way with exponential security. (That is, no 2^{n^k}-time-bounded algorithm with n^k bits of nonuniform advice inverts f on more than an exponentially small set of inputs.) 2. If DTIME(2^n) ``separates all BPP pairs,\u27\u27 then there is a (polynomial time computable) pseudorandom generator that passes all probabilistic polynomial-time statistical tests. (This result is a partial converse of Yao, Boppana, and Hirschfeld\u27s theorem, that the existence of pseudorandom generators passing all polynomial-size circuit statistical tests implies that BPP\subset DTIME(2^{n^epsilon}) for all epsilon\u3e0.) Such consequences are not known to follow from the weaker hypothesis that P \u3c\u3e NP

One-Way Functions and Balanced NP

The existence of cryptographically secure one-way functions is related to the measure of a subclass of NP. This subclass, called BNP ( balanced NP ), contains 3SAT and other standard NP problems. The hypothesis that BNP is not a subset of P is equivalent to the P not equal to NP conjecture. A stronger hypothesis, that BNP is not a measure 0 subset of E_2 = DTIME(2^polynomial) is shown to have the following two consequences. 1. For every k, there is a polynomial time computable, honest function f that is (2^{n^k}/n^k)-one-way with exponential security. (That is, no 2^{n^k}-time-bounded algorithm with n^k bits of nonuniform advice inverts f on more than an exponentially small set of inputs.) 2. If DTIME(2^n) separates all BPP pairs, then there is a (polynomial time computable) pseudorandom generator that passes all probabilistic polynomial-time statistical tests. (This result is a partial converse of Yao, Boppana, and Hirschfeld\u27s theorem, that the existence of pseudorandom generators passing all polynomial-size circuit statistical tests implies that BPP is a subset of DTIME(2^{n^epsilon}) for all epsilon\u3e0.) Such consequences are not known to follow from the weaker hypothesis that P is not equal to NP

Complexity Theory

Computational Complexity Theory is the mathematical study of the intrinsic power and limitations of computational resources like time, space, or randomness. The current workshop focused on recent developments in various sub-areas including arithmetic complexity, Boolean complexity, communication complexity, cryptography, probabilistic proof systems, pseudorandomness, and quantum computation. Many of the developements are related to diverse mathematical fields such as algebraic geometry, combinatorial number theory, probability theory, quantum mechanics, representation theory, and the theory of error-correcting codes