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Cryptanalysis of RSA-type cryptosystems based on Lucas sequences, Gaussian integers and elliptic curves

Abstract : In 1995, Kuwakado, Koyama and Tsuruoka presented a new RSA-type scheme based on singular cubic curves y^2 ≡ x^3 + bx^2 (mod 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 = P Q 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 based on Lucas sequences. In the three schemes, the public exponent e is an integer satisfying the key equation ed − k (p^2 − 1)(q^2 − 1 )= 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 experiments demonstrate that for a 1024-bit modulus, our method works for values of d of up to 520 bits. We also examine the effect of dropping the usual assumption that p and q have the same bit size.
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Contributor : Abderrahmane Nitaj <>
Submitted on : Sunday, October 20, 2019 - 12:49:51 AM
Last modification on : Monday, April 27, 2020 - 4:14:03 PM
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Martin Bunder, Abderrahmane Nitaj, Willy Susilo, Joseph Tonien. Cryptanalysis of RSA-type cryptosystems based on Lucas sequences, Gaussian integers and elliptic curves. Journal of information security and applications, 2018, ⟨10.1016/j.jisa.2018.04.006⟩. ⟨hal-02320970⟩

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