Usually, a proof of a theorem contains more knowledge than the mere fact that the theorem is true. For instance, to prove that a graph is Hamiltonian it suffices to exhibit a Hamiltonian tour in it; however, this seems to contain more knowledge than the single bit Hamiltonian/non-Hamiltonian.In this paper a computational complexity theory of the “knowledge” contained in a proof is developed. Zero-knowledge proofs are defined as those proofs that convey no additional knowledge other than the correctness of the proposition in question. Examples of zero-knowledge proof systems are given for the languages of quadratic residuosity and 'quadratic nonresiduosity. These are the first examples of zero-knowledge proofs for languages not known to be efficiently recognizable.

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The knowledge complexity of interactive proof-systems

We develop an automated approach for designing matrix multiplication algorithms based on constructions similar to the Coppersmith-Winograd construction. Using this approach we obtain a new improved bound on the matrix multiplication exponent ω

/pdf/multiplying-matrices-faster-than-coppersmith-winograd-1xvjqg5bxt.pdf

Multiplying matrices faster than coppersmith-winograd

This paper presents a method to analyze the powers of a given trilinear form (a special kind of algebraic constructions also called a tensor) and obtain upper bounds on the asymptotic complexity of matrix multiplication. Compared with existing approaches, this method is based on convex optimization, and thus has polynomial-time complexity. As an application, we use this method to study powers of the construction given by Coppersmith and Winograd [Journal of Symbolic Computation, 1990] and obtain the upper bound $\omega<2.3728639$ on the exponent of square matrix multiplication, which slightly improves the best known upper bound.

Powers of Tensors and Fast Matrix Multiplication

Computer algebra systems are now ubiquitous in all areas of science and engineering. This highly successful textbook, widely regarded as the “bible of computer algebra”, gives a thorough introduction to the algorithmic basis of the mathematical engine in computer algebra systems. Designed to accompany oneor two-semester courses for advanced undergraduate or graduate students in computer science or mathematics, its comprehensiveness and reliability has also made it an essential reference for professionals in the area. Special features include: detailed study of algorithms including time analysis; implementation reports on several topics; complete proofs of the mathematical underpinnings; and a wide variety of applications (among others, in chemistry, coding theory, cryptography, computational logic, and the design of calendars and musical scales). A great deal of historical information and illustration enlivens the text. In this third edition, errors have been corrected and much of the Fast Euclidean Algorithm chapter has been renovated.

Modern computer algebra

Recently several results appeared that show significant reduction in time for matrix multiplication, singular value decomposition as well as linear (\ell_ 2) regression, all based on data dependent random sampling. Our key idea is that low dimensional embeddings can be used to eliminate data dependence and provide more versatile, linear time pass efficient matrix computation. Our main contribution is summarized as follows. --Independent of the recent results of Har-Peled and of Deshpande and Vempala, one of the first -- and to the best of our knowledge the most efficient -- relative error (1 + \in) \parallel A - A_k \parallel _F approximation algorithms for the singular value decomposition of an m ? n matrix A with M non-zero entries that requires 2 passes over the data and runs in time O\left( {\left( {M(\frac{k} { \in } + k\log k) + (n + m)(\frac{k} { \in } + k\log k)^2 } \right)\log \frac{1} {\delta }} \right) --The first o(nd^{2}) time (1 + \in) relative error approximation algorithm for n ? d linear (\ell_2) regression. --A matrix multiplication and norm approximation algorithm that easily applies to implicitly given matrices and can be used as a black box probability boosting tool.

Improved Approximation Algorithms for Large Matrices via Random Projections

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 one of these to be quasipolynomial in the optimal. Our expanders have a short and self-contained description and analysis, based on the ideas underlying the recent list-decodable error-correcting codes of Parvaresh and Vardy [2005].Our expanders can be interpreted as near-optimal “randomness condensers,” that reduce the task of extracting randomness from sources of arbitrary min-entropy rate to extracting randomness from sources of min-entropy rate arbitrarily close to 1, which is a much easier task. Using this connection, we obtain a new, self-contained construction of randomness extractors that is optimal up to constant factors, while being much simpler than the previous construction of Lu et al. [2003] and improving upon it when the error parameter is small (e.g., 1/poly(n)).

/pdf/unbalanced-expanders-and-randomness-extractors-from-irx83xv27b.pdf

Unbalanced expanders and randomness extractors from Parvaresh--Vardy codes

We further develop the group-theoretic approach to fast matrix multiplication introduced by Cohn and Umans, and for the first time use it to derive algorithms asymptotically faster than the standard algorithm. We describe several families of wreath product groups that achieve matrix multiplication exponent less than 3, the asymptotically fastest of which achieves exponent 2.41. We present two conjectures regarding specific improvements, one combinatorial and the other algebraic. Either one would imply that the exponent of matrix multiplication is 2.

/pdf/group-theoretic-algorithms-for-matrix-multiplication-y6m6jm1u4b.pdf

Group-theoretic algorithms for matrix multiplication

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 that produces good extractors for all min-entropies Our construction is algebraic and builds on a new polynomial-based approach introduced by Ta-Shma et al [2001b] Using our improvements, we obtain, for example, an extractor with output length m = k/(log n)O(1/α) and seed length (1 p α)log n for an arbitrary 0

Simple extractors for all min-entropies and a new pseudorandom generator

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
$log q ge n$, it matches the asymptotic running time of the best
known algorithms.

The improvements come from new algorithms for modular composition
of degree $n$ univariate polynomials, which is the asymptotic
bottleneck in fast algorithms for factoring polynomials over
finite fields. The best previous algorithms for modular
composition use $O(n^{(omega + 1)/2})$ field operations, where
$omega$ is the exponent of matrix multiplication (Brent & Kung
(1978)), with a slight improvement in the exponent achieved by
employing fast rectangular matrix multiplication (Huang & Pan
(1997)).

We show that modular composition and multipoint evaluation of
multivariate polynomials are essentially equivalent, in the sense
that an algorithm for one achieving exponent $alpha$ implies an
algorithm for the other with exponent $alpha + o(1)$, and vice
versa. We then give two new algorithms that solve the problem
optimally (up to lower order terms): an algebraic algorithm for
fields of characteristic at most $n^{o(1)}$, and a
nonalgebraic algorithm that works in arbitrary characteristic.
The latter algorithm works by lifting to characteristic 0,
applying a small number of rounds of {em multimodular reduction},
and finishing with a small number of multidimensional FFTs. The
final evaluations are reconstructed using the Chinese Remainder
Theorem. As a bonus, this algorithm produces a very efficient data
structure supporting polynomial evaluation queries, which is of
independent interest.

Our algorithms use techniques which are commonly employed in
practice, so they may be competitive for real problem sizes. This
contrasts with all previous subquadratic algorithsm for these
problems, which rely on fast matrix multiplication.

This is joint work with Kiran Kedlaya.

Fast polynomial factorization and modular composition.

We obtain randomized algorithms for factoring degree $n$ univariate polynomials over $\mathbb{F}_q$ requiring $O(n^{1.5 + o(1)}\,{\rm log}^{1+o(1)} q+ n^{1 + o(1)}\,{\rm log}^{2+o(1)} q)$ bit operations. When ${\rm log}\, q < n$, this is asymptotically faster than the best previous algorithms [J. von zur Gathen and V. Shoup, Comput. Complexity, 2 (1992), pp. 187-224; E. Kaltofen and V. Shoup, Math. Comp., 67 (1998), pp. 1179-1197]; for ${\rm log}\, q \ge n$, it matches the asymptotic running time of the best known algorithms. The improvements come from new algorithms for modular composition of degree $n$ univariate polynomials, which is the asymptotic bottleneck in fast algorithms for factoring polynomials over finite fields. The best previous algorithms for modular composition use $O(n^{(\omega + 1)/2})$ field operations, where $\omega$ is the exponent of matrix multiplication [R. P. Brent and H. T. Kung, J. Assoc. Comput. Mach., 25 (1978), pp. 581-595], with a slight improvement in the exponent achieved by employing fast rectangular matrix multiplication [X. Huang and V. Y. Pan, J. Complexity, 14 (1998), pp. 257-299]. We show that modular composition and multipoint evaluation of multivariate polynomials are essentially equivalent, in the sense that an algorithm for one achieving exponent $\alpha$ implies an algorithm for the other with exponent $\alpha + o(1)$, and vice versa. We then give two new algorithms that solve the problem near-optimally: an algebraic algorithm for fields of characteristic at most $n^{o(1)}$, and a nonalgebraic algorithm that works in arbitrary characteristic. The latter algorithm works by lifting to characteristic 0, applying a small number of rounds of multimodular reduction, and finishing with a small number of multidimensional FFTs. The final evaluations are reconstructed using the Chinese remainder theorem. As a bonus, this algorithm produces a very efficient data structure supporting polynomial evaluation queries, which is of independent interest. Our algorithms use techniques that are commonly employed in practice, in contrast to all previous subquadratic algorithms for these problems, which relied on fast matrix multiplication.

/pdf/fast-polynomial-factorization-and-modular-composition-3v05yipo37.pdf

Christopher Umans

Papers

Unbalanced expanders and randomness extractors from Parvaresh--Vardy codes

Group-theoretic algorithms for matrix multiplication

Simple extractors for all min-entropies and a new pseudorandom generator

Fast polynomial factorization and modular composition.

Fast Polynomial Factorization and Modular Composition