Private Visual Share-Homomorphic Computation and Randomness Reduction in Visual Cryptography

Secure computation through non standard methods, suitable for users who have to perform the computation without the aid of a computer, or for settings in which the degree of trustworthiness of the hardware and software equipments is very low, are an interesting, very challenging and quite unexplored research topic. In this paper we put forward a collection of ideas and some techniques which could be useful in order to make some progress in designing protocols with such properties. Our contribution is twofold: we explore the power of visual cryptography as a computing tool, exploiting alternative uses and share manipulations, and we address the central issue of randomness reduction in visual schemes, by showing a strict relation with existing results in secure multiparty computation. More specifically, we prove that: by properly defining operations on the shares, we show that visual shares are homomorphic with respect to some functions f. More precisely, in the two-party case, each user, by applying to his two shares a(i), b(i) of the secrets a, b the operation, gets a share g(i)(a(i), b(i)), i = 1, 2, such that the superposition of g(1)(a(1), b(1)) and g(2)(a(2), b(2)) visually provides, applying the standard Naor and Shamir superposition reconstruction strategy, the value of the function f; we link our analysis to a general known result on private two-party computation, and we classify all the boolean functions of two input bits which admit homomorphic visual share computation; we prove that by encoding pixels in groups, instead of encoding each pixel independently, and exploiting dependencies, some randomness can be saved if and only if the pixel dependencies can be expressed through some specific boolean functions. For example, given three pixels, if the third one is the and or the or of the first two, randomness reduction is impossible, while if it is the xor of the first two, randomness reduction can be achieved.