Dual protocols for private multi-party matrix multiplication and trust computations

Abstract

This paper deals with distributed matrix multiplication. Each player owns only one row of both matrices and wishes to learn about one distinct row of the product matrix, without revealing its input to the other players. We first improve on a weighted average protocol, in order to securely compute a dot-product with a quadratic volume of communications and linear number of rounds. We also propose two dual protocols with five communication rounds, using a Paillier-like underlying homomorphic public key cryptosystem, which is secure in the semihonest model or secure with high probability in the malicious adversary model. Using cryptographic protocol verification tools, we are able to check the security of both protocols and provide a countermeasure for each attack found by the tools. We also give a generic randomization method to avoid collusion attacks. As an application, we show that this protocol enables a distributed and secure evaluation of trust relationships in a network, for a large class of trust evaluation schemes.

Secure Multiparty Matrix Multiplication Based on Strassen-Winograd Algorithm

“…We consider the setting where the two input matrices A and B have dimension N × N and each of the N players stores one row of A and the corresponding row of B and learns the corresponding row of C = A × B. In this setting, the YTP-SS Algorithm [11,Algorithm 15] can compute C by encrypting the rows of A only and then relying on homomorphic multiplications of encrypted coefficients of A by plain coefficients of B.…”

“…For instance, for a product of dimension 12, with base case dimension b = 3, this gives; L A = L B = L C = (1,2,0,4,5,3,7,8,6,11,9,10) and K A = K B = K C = (0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11).…”

“…The matrix multiplication algorithm using the secure dot-product protocol YTP-SS [11,Algorithm 15] is secure against semi-honest adversaries over insecure communication channels. In order to analyze the difference with our proposition, Protocol 7 MP-SW, we extract here the core of the former protocol, i.e., without the securization of the channel (that is we remove the protection of the players private elements by random values, and the final communications to derandomize the results).…”

“…Some of them are for two parties only and most of the others are generic and transform programs into circuits or use oblivious transfer [7,26,6,16,23]. For instance the symmetric system solving phase of the Linreg-MPC software is reported in [12] to take about 45 minutes for n = 200, while, in [11], a secure multiparty specific algorithm, YTP-SS, was developed for matrix multiplication requires about a hundred seconds to perform an n = 200 matrix multiplication. These timings, however, do not take into account communications, but for multiparty matrix multiplication, the number of communications and the number of operations should be within the same order of magnitude.…”

“…Finally, Strassen-Winograd algorithm involves numerous additions and subtractions on parts of the A and B matrices that are held by different players. Security concerns require then that these entries should be encrypted from the start, contrarily to [11]. As a consequence, the classical matrix multiplication can no longer be used as stated in the latter reference, even for the base case of the recursive algorithm.…”