An extremely small and efficient identification scheme

We present a new identification scheme which is based on Legendre symbols modulo a certain hidden prime and which is naturally suited for low power, low memory applications.7 page(s

Variations of Diffie-Hellman problem

Abstract. This paper studies various computational and decisional Diffie-Hellman problems by providing reductions among them in the high granularity setting. We show that all three variations of computational Diffie-Hellman problem: square Diffie-Hellman problem, inverse Diffie-Hellman problem and divisible Diffie-Hellman problem, are equivalent with optimal reduction. Also, we are considering variations of the decisional Diffie-Hellman problem in single sample and polynomial samples settings, and we are able to show that all variations are equivalent except for the argument DDH ⇐ SDDH. We are not able to prove or disprove this statement, thus leave an interesting open problem. Keywords: Diffie-Hellman problem, Square Diffie-Hellman problem, Inverse Diffie-Hellman problem, Divisible Diffie-Hellman proble

The Group of Signed Quadratic Residues and Applications

We consider the cryptographic group of Signed Quadratic Residues. This group is particularly useful for cryptography since it is a “gap-group, ” in which the computational problem (i.e., computing square roots) is as hard as factoring, while the corresponding decisional problem (i.e., recognizing signed quadratic residues) is easy. We are able to show that under the factoring assumption, the Strong Diffie-Hellman assumption over the signed quadratic residues holds. That is, in this group the Diffie-Hellman problem is hard, even in the presence of a Decisional Diffie-Hellman oracle. We demonstrate the usefulness of our results by applying them to the Hybrid ElGamal encryption scheme (aka Diffie-Hellman integrated encryption scheme — DHIES). Concretely, we consider the security of the scheme when instantiated over the group of signed quadratic residues. It is known that, in the random oracle model, the scheme is chosen-ciphertext (CCA) secure under the Strong Diffie-Hellman assumption and hence, by our results, under the standard factoring assumption. We show that furthermore, in the standard model, Hybrid ElGamal is CCA secure under the higher residuosity assumption, given that the used hash function is four-wise independent. The latter result is obtained using the recent “randomness extraction framework ” for hash proof systems. Keywords: Public-key encryption, chosen-ciphertext security, Hybrid ElGamal/DHIE