Generalized Scheme For Fractal Based Digital Signature (GFDS).

This paper describes a new development in the cryptographic digital signature scheme based on Mandelbrot and Julia fractal sets. Recently it has been shown that it is possible to have digital signature scheme based on fractal due to the strong connection between the Mandelbrot and Julia fractal sets

Unique Rabin-Williams Signature Scheme Decryption

Abstract. The extremely efficient Rabin-Williams signature scheme relies on decryption of a quadratic equation in order to retrieve the original message. Customarily, square roots are found using the Chinese Remainder Theorem. This can be done in polynomial time, but generally produces four options for the correct message which must be analyzed to determine the correct one. This paper resolves the problem of efficient deterministic decryption to the correct message modulo $p^2q$ by establishing conditions on the primes $p$ and $q$ as well as on any legitimate message. We do this using the CRT modulo pq to find four roots. We show that the correct root (initial message) is the only one of these four which is in our allowed message set (it is in fact the smallest of the four integers) and which satisfies a quadratic equation modulo $p^2q$; no additional work is required to eliminate the others. As a result, we propose what we believe is now the most efficient version of R-W signature scheme decryption

On the Lossiness of the Rabin Trapdoor Function

Lossy trapdoor functions, introduced by Peikert and Waters (STOC~\u2708), are functions that can be generated in two indistinguishable ways: either the function is injective, and there is a trapdoor to invert it, or the function is lossy, meaning that the size of its range is strictly smaller than the size of its domain. Kakvi and Kiltz (EUROCRYPT 2012) proved that the Full Domain Hash signature scheme based on a lossy trapdoor function has a \emph{tight} security reduction from the lossiness of the trapdoor function. Since Kiltz, O\u27Neill, and Smith (CRYPTO 2010) showed that the RSA trapdoor function is lossy under the $\Phi$-Hiding assumption of Cachin, Micali, and Stadler (EUROCRYPT~\u2799), this implies that the RSA Full Domain Hash signature scheme has a \emph{tight} security reduction from the $\Phi$-Hiding assumption (for public exponents $e<N^{1/4}$). In this work, we consider the Rabin trapdoor function, \emph{i.e.} modular squaring over $\mathbb{Z}_{N}^*$. We show that when adequately restricting its domain (either to the set $\mathbb{QR}_{N}$ of quadratic residues, or to $(\mathbb{J}_{N})^+$, the set of positive integers $1\le x\le(N-1)/2$ with Jacobi symbol +1) the Rabin trapdoor function is lossy, the injective mode corresponding to Blum integers $N=pq$ with $p,q\equiv 3\bmod 4$, and the lossy mode corresponding to what we call pseudo-Blum integers $N=pq$ with $p,q\equiv 1 \bmod 4$. This lossiness result holds under a natural extension of the $\Phi$-Hiding assumption to the case $e=2$ that we call the 2-$\Phi/4$-Hiding assumption. We then use this result to prove that deterministic variants of Rabin-Williams Full Domain Hash signatures have a tight reduction from the 2-$\Phi$/4-Hiding assumption. We also show that these schemes are unlikely to have a tight reduction from the factorization problem by extending a previous ``meta-reduction\u27\u27 result by Coron (EUROCRYPT 2002), later corrected by Kakvi and Kiltz (EUROCRYPT 2012). These two results therefore answer one of the main questions left open by Bernstein (EUROCRYPT 2008) in his work on Rabin-Williams signatures

Critical Perspectives on Provable Security: Fifteen Years of Another Look Papers

We give an overview of our critiques of “proofs” of security and a guide to our papers on the subject that have appeared over the past decade and a half. We also provide numerous additional examples and a few updates and errata