A new generalization of the KMOV cryptosystem

Abstract

The KMOV scheme is a public key cryptosystem based on an RSA modulus n = pq where p and q are large prime numbers with p ≡ q ≡ 2 (mod 3). It uses the points of an elliptic curve with equation y 2 ≡ x 3 + b (mod n). In this paper, we propose a generalization of the KMOV cryptosystem with a prime power modulus of the form n = p r q s and study its resistance to the known attacks.

Classical Attacks on a Variant of the RSA Cryptosystem

“…Astonishingly, this is roughly the same bound than the classical bound obtained by Wiener's method for standard RSA. Similarly, we show that the method of Boneh and Durfee can be applied if δ < 7 3 − 2 3 √ 3α + 1. When e ≈ N 2 , the bound reduces to d < N 0.5694 .…”

A new public key cryptosystem based on Edwards curves

“…We have studied the arithmetical properties of the twisted Edwards curves on the finite field Z/pZ and generalized them to the rings Z/p r Z and Z/p r q s Z. Using these properties, we have proposed a new public key scheme which can be seen as a generalization of two former public key schemes: the KMOV cryptosystem [12] with an RSA modulus and its generalization to a prime power RSA modulus [5].…”

“…This implies that E a,d,p has no singular points. Thus, using the generalized Hensel Lemma (see [5]), to the polynomial f (x, y), we deduce that #E a,d,p r = p r−1 #E a,d,p .…”

“…In [12], Koyama et al introduced a variant of the RSA cryptosystem based on an RSA modulus n = pq and an elliptic curve with equation y 2 ≡ x 3 + b (mod n). This was recently extended by Boudabra and Nitaj [5] using a prime power modulus n = p r q s .…”

“…Then, using the arithmetic properties of the twisted Edwards curves on the ring Z/p r q s Z, we propose a new public key scheme and study its efficiency and its security. The new scheme generalizes two former schemes, namely the KMOV cryptosystem [12] with a modulus of the form n = pq and an elliptic curve with equation y 2 ≡ x 3 + b (mod n) and its extension to a prime power RSA modulus n = p r q s with a similar equation [5]. The new scheme uses a prime power RSA modulus n = p r q s and a twisted Edwards curve with equation…”