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Published in:
2016 | OriginalPaper | Chapter
A New Attack on Three Variants of the RSA Cryptosystem
Authors : Martin Bunder, Abderrahmane Nitaj, Willy Susilo, Joseph Tonien
Published in: Information Security and Privacy
Publisher: Springer International Publishing
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Abstract
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.