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

机译：基于Lucas序列，高斯整数和椭圆曲线的RSA类型密码系统的密码分析

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In this paper, we apply the continued fraction method to launch an attack on the three RSA-type cryptosystems when the private exponent d is sufficiently small. The first cryptosystem, proposed by Kuwakado, Koyama and Tsuruoka in 1995, is a scheme based on singular cubic curves y~2 = x~3+bx~2 (mod N) where N = pq is an RSA modulus. The second cryptosystem, proposed by Elkamchouchi, Elshenawy and Shaban in 2002, is 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. The third cryptosystem, proposed by Castagnos in 2007, is a scheme over quadratic field quotients with an RSA modulus N = pq based on Lucas sequences. In the three cryptosystems, the public exponent e is an integer satisfying the key equation ed - k(p~2 - 1)(q~2 - 1) = 1. Our attack is applicable to primes p and q of arbitrary sizes and we do not require the usual assumption that p and q have the same bit size. Thus, this is an extension of our recent result presented at ACISP 2016 conference. Our experiments demonstrate that for a 513-bit prime p and 511-bit prime q, our method works for values of d of up to 520 bits.
机译：在本文中，当私有指数d足够小时，我们应用连续分数法对三个RSA型密码系统发起攻击。 Kuwakado，Koyama和Tsuruoka在1995年提出的第一个密码系统是一种基于奇异三次曲线y〜2 = x〜3 + bx〜2（mod N）的方案，其中N = pq是RSA模数。 Elkamchouchi，Elshenawy和Shaban在2002年提出的第二种密码系统是使用模数N = PQ将RSA方案扩展到高斯整数域，其中P和Q是高斯素数，使得p = | P |和q = | Q |是普通素数。 Castagnos在2007年提出的第三种密码系统是一种基于Lucas序列的具有RSA模数N = pq的二次场商的方案。在这三个密码系统中，公共指数e是一个满足密钥方程ed-k（p〜2-1-（q〜2-1-1）= 1的整数）。我们的攻击适用于任意大小的素数p和q不需要通常假设p和q具有相同的位大小。因此，这是我们在ACISP 2016大会上提出的最新结果的扩展。我们的实验表明，对于513位素数p和511位素数q，我们的方法适用于d值高达520位的情况。

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