A New Attack on Three Variants of the RSA Cryptosystem
In 1995, Kuwakado, Koyama and Tsuruoka presented a new RSA type scheme based on singular cubic curves $y^2\equiv x^3+bx^2\pmod N$ where $N=pq$ is an RSA modulus. Then, in 2002, Elkamchouchi, Elshenawy and Shaban introduced an extension of the RSA scheme to the field of Gaussian integers using a modulus $N=PQ$ where P and Q are Gaussian primes such that $p=|P| and $q=|Q| are ordinary primes. Later, in 2007, Castagnos proposed a scheme over quadratic field quotients with an RSA modulus $N=pq$ In the three schemes, the public exponent e is an integer satisfying the key equation $ed k\left p^2 1\right \left q^2 1\right =1$ In this paper, we apply the continued fraction method to launch an attack on the three schemes when the private exponent d is sufficiently small. Our attack can be considered as an extension of the famous Wiener attack on the RSA.
ACISP