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The <i>exponent of matrix multiplication</i> is the smallest real number ω such that for all ε>0, <i>O</i>(n<sup>ω+ε</sup>) arithmetic operations suffice to multiply two <i>n×n</i> matrices. The standard algorithm for matrix multiplication shows that ω≤3. Strassen's remarkable result [5] shows that ω≤2.81,… (More)

A “randomness extractor” is an algorithm that given a sample from a distribution with sufficiently high min-entropy and a short random seed produces an output that is statistically indistinguishable from uniform. (Min-entropy is a measure of the amount of randomness in a distribution.) We present a simple, self-contained extractor construction… (More)

We give an improved explicit construction of highly unbalanced bipartite expander graphs with expansion arbitrarily close to the degree (which is polylogarithmic in the number of vertices). Both the degree and the number of right-hand vertices are polynomially close to optimal, whereas the previous constructions of Ta-Shma et al. [2007] required at least… (More)

We develop a new, group-theoretic approach to bounding the exponent of matrix multiplication. There are two components to this approach: (1) identifying groups G that admit a certain type of embedding of matrix multiplication into the group algebra C[G], and (2) controlling the dimensions of the irreducible representations of such groups. We present… (More)

We obtain randomized algorithms for factoring degree n univariate polynomials over F q requiring O(n 1.5+o(1) log 1+o(1) q + n 1+o(1) log 2+o(1) q) bit operations. When log q < n, this is asymptotically faster than the best previous algorithms (von zur Gathen & Shoup (1992) and Kaltofen & Shoup (1998)); for log q ≥ n, it matches the asymptotic running time… (More)

We study computational procedures that use both randomness and nondeterminism. Examples are Arthur-Merlin games and approximate counting and sampling of NP-witnesses. The goal of this paper is to derandomize such procedures under the weakest possible assumptions. Our main technical contribution allows one to " boost " a given hardness assumption. One… (More)

(MATH) We construct the first pseudo-random generators with logarithmic seed length that convert <i>s</i> bits of hardness into <i>s</i><sup>Ω(1)</sup> bits of 2-sided pseudo-randomness <i>for any s</i>}. This improves [8] and gives a direct proof of the optimal hardness vs. randomness tradeoff in [15]. A key element in our construction is an… (More)

An extractor is a procedure which extracts randomness from a detective random source using a few additional random bits. Explicit extractor constructions have numerous applications and obtaining such constructions is an important derandomization goal. Trevisan recently introduced an elegant extractor construction, but the number of truly random bits… (More)

We study the complexity of solving succinct zero-sum games, i.e., the games whose payoff matrix M is given implicitly by a Boolean circuit C such that M (i, j) = C(i, j). We complement the known EXP-hardness of computing the exact value of a succinct zero-sum game by several results on approximating the value. (1) We prove that approximating the value of a… (More)

We give an algorithm for modular composition of degree n univariate polynomials over a finite field F q requiring n 1+o(1) log 1+o(1) q bit operations; this had earlier been achieved in characteristic n o(1) by Umans (2008). As an application, we obtain a randomized algorithm for factoring degree n polynomials over F q requiring (n 1.5+o(1) + n 1+o(1) log… (More)