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Tomas Sander

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2 papers

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FOCS Conference 1999 Conference Paper

Non-Interactive CryptoComputing For NC 1

Tomas Sander
Adam L. Young
Moti Yung

The area of "computing with encrypted data" has been studied by numerous authors in the past twenty years since it is fundamental to understanding properties of encryption and it has many practical applications. The related fundamental area of "secure function evaluation" has been studied since the mid 80's. In its basic two-party case, two parties (Alice and Bob) evaluate a known circuit over private inputs (or a private input and a private circuit). Much attention has been paid to the important issue of minimizing rounds of computation in this model. Namely, the number of communication rounds in which Alice and Bob need to engage in to evaluate a circuit on encrypted data securely. Advancements in these areas have been recognized as open problems and have remained open for a number of years. In this paper we give a one round, and thus round optimal, protocol for secure evaluation of circuits which is in polynomial time for NC/sup 1/ circuits. The protocol involves an input party sending encrypted input to a second party, a cryptocomputer, which evaluates the circuit (or a known circuit over its additional private input) non-interactively, securely and obliviously, and provides the output to the input party without learning it. This improves on previous (general) results that are specialized to the case of NC/sup 1/ circuits and require a constant number of communication rounds. We further suggest applications to network and mobile computing.

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FOCS Conference 1997 Conference Paper

Deciding Properties of Polynomials Without Factoring

Tomas Sander
Mohammad Amin Shokrollahi

The polynomial time algorithm of Lenstra, Lenstra, and Lovasz (1982) for factoring integer polynomials and variants thereof have been widely used to show that various computational problems in number theory have polynomial time solutions. Among them is the problem of factoring polynomials over algebraic number fields, which is used itself as a major subroutine for several other algorithms. Although a theoretical breakthrough, algorithms based on factorization of polynomials are notoriously slow and hard to implement, with running times ranging between O(n/sup 12/) and O(n/sup 18/) depending on which variant of the lattice basis reduction is used. Here, n is an upper bound for the maximum of the degrees and the bit-lengths of the coefficients of the polynomials involved. On the other hand, in many situations one does not need the full power of factorization, so one may ask whether there exist faster algorithms in these cases. In this paper we develop more efficient Monte Carlo algorithms to decide certain properties of roots of integer polynomials, without factoring them. Such problems arise, e. g. , when solving systems of algebraic equations. Our methods applied to this situation thus give information about the solutions of such systems of equations.

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