Fast Secure Comparison for Medium-Sized Integers and Its Application in Binarized Neural Networks

Abstract

In 1994, Feige, Kilian, and Naor proposed a simple protocol for secure 3-way comparison of integers a and b from the range [0, 2]. Their observation is that for p = 7, the Legendre symbol (x | p) coincides with the sign of x for x = a−b ∈ [−2, 2], thus reducing secure comparison to secure evaluation of the Legendre symbol. More recently, in 2011, Yu generalized this idea to handle secure comparisons for integers from substantially larger ranges [0, d], essentially by searching for primes for which the Legendre symbol coincides with the sign function on [−d, d]. In this paper, we present new comparison protocols based on the Legendre symbol that additionally employ some form of error correction. We relax the prime search by requiring that the Legendre symbol encodes the sign function in a noisy fashion only. Practically, we use the majority vote over a window of 2k + 1 adjacent Legendre symbols, for small positive integers k. Our technique significantly increases the comparison range: e.g., for a modulus of 60 bits, d increases by a factor of 2.9 (for k = 1) and 5.4 (for k = 2) respectively. We give a practical method to find primes with suitable noisy encodings. We demonstrate the practical relevance of our comparison protocol by applying it in a secure neural network classifier for the MNIST dataset. Concretely, we discuss a secure multiparty computation based on the binarized multi-layer perceptron of Hubara et al., using our comparison for the second and third layers.

Blockchain-based Practical Multi-agent Secure Comparison and its Application in Auctions

“…These agents also know one of the values, i.e., x or y. This implies, from Corollary 3.1, that the probability with which these agents can guess the value of x or y from X + Y is 1 2 d . Thus in SCP, the threshold of the probability with which a party can guess the value of x (y) is 1 2 d , which is negligible in d. ZKP in SCP.…”

“…To the best of our knowledge, the existing protocols comprise one or more of (i) garbled circuits, (ii) partialhomomorphic encryption, and (iii) continuous involvement of the parties during execution. Note that, [1] is limited as it is only applicable for integers in the range of ≈ 2 60 ; whereas SCP works for any range. Consequently, these protocols can not be adapted towards designing lightweight and secure AI applications.…”

“…To securely deploy these AI applications over blockchain, we present SCP over the Ethereum network 1 . To the best of our knowledge, we are the first to introduce a dedicated and verifiable solution, 1 SCP may also be deployed using computationally expensive secure channels like Tor [10]. By coupling blockchain with an asymmetric encryption scheme, we ensure efficient pseudo-anonymous communication.…”

Three-Party Integer Comparison and Applications

“…The communications and computations are inefficient requiring many rounds of interaction. In contrast to bitwise comparisons, more efficient protocols were given in [7,8,9,10] to compare multiple bits simultaneously with a single ciphertext by a somewhat homomorphic encryption.…”