Abstract
This work deals with the communication complexity of secure multi-party protocols for linear algebra problems. In our model, complexity is measured in terms of the number of secure multiplications required and protocols terminate within a constant number of rounds of communication.
Previous work by Cramer and Damgård proposes secure protocols for solving systems Ax = b of m linear equations in n variables over a finite field, with m ≤ n. The complexity of those protocols is n 5.
We show a new upper bound of m 4 + n 2 m secure multiplications for this problem, which is clearly asymptotically smaller. Our main point, however, is that the advantage can be substantial in case m is much smaller than n. Indeed, if \(m={\sqrt{n}}\), for example, the complexity goes down from n 5 to n 2.5.
Our secure protocols rely on some recent advances concerning the computation of the Moore-Penrose pseudo-inverse of matrices over fields of positive characteristic. These computations are based on the evaluation of a certain characteristic polynomial, in combination with variations on a well-known technique due to Mulmuley that helps to control the effects of non-zero characteristic. We also introduce a new method for secure polynomial evaluation that exploits properties of Chebychev polynomials, as well as a new secure protocol for computing the characteristic polynomial of a matrix based on Leverrier’s lemma that exploits this new method.
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Keywords
- Characteristic Polynomial
- Chebyshev Polynomial
- Secure Protocol
- Secure Computation
- Multiplication Protocol
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Cramer, R., Kiltz, E., Padró, C. (2007). A Note on Secure Computation of the Moore-Penrose Pseudoinverse and Its Application to Secure Linear Algebra. In: Menezes, A. (eds) Advances in Cryptology - CRYPTO 2007. CRYPTO 2007. Lecture Notes in Computer Science, vol 4622. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-74143-5_34
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