Fast secure comparison for medium-sized integers and its application in binarized neural networks

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 2.8 (for k=1) and 3.8 (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.</p

Fast secure comparison for medium-sized integers and its application in binarized neural networks

\u3cp\u3eIn 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 (Formula Presented), the Legendre symbol (Formula Presented) coincides with the sign of x for (Formula Presented), 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 (Formula Presented). 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 (Formula Presented) 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 2.8 (for (Formula Presented)) and 3.8 (for (Formula Presented)) 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.\u3c/p\u3

Information leakage of continuous-source zero secrecy leakage helper data schemes

A Helper Data Scheme is a cryptographic primitive that extracts a high-entropy noise-free string from noisy data. Helper Data Schemes are used for privacy-preserving databases and for Physical Unclonable Functions. We refine the theory of Helper Data schemes with Zero Secrecy Leakage (ZSL), i.e. the mutual information between the helper data and the extracted secret is zero. We prove that ZSL necessitates particular properties of the helper data generating function, which also allows us to show the existence of `Sibling Points'. In the special case that our generated secret is uniformly distributed (Fuzzy Extractors) our results coincide with the continuum limit of a recent construction by Verbiskiy et al. Yet our results cover secure sketches as well. Moreover we present an optimal reconstruction algorithm for this scheme, that not only provides the lowest possible reconstruction error rate but also yields an attractive, simple implementation of the verification. Further, we introduce Diagnostic Category Leakage (DCL), which quantifies what an attacker can infer from helper data about a particular medical indication of the enrolled user, or reversely what probabilistic knowledge of a diagnose can leak about the secret. If the attacker has a priori knowledge about the enrolled user (medical indications, race, gender), then the ZSL property does not guarantee that there is no secrecy leakage from the helper data. However, this effect is typically very small