Weak Composite Diffie-Hellman

In1985, Shmuley proposed a theorem about intractability of Composite Diffie-Hellman. The theorem of Shmuley may be paraphrased as saying that if there exist a probabilistic polynomial time oracle machine which solves the Diffie-Hellman modulo an RSA-number with odd-order bases then there exist a probabilistic algorithm which factors the modulo. In the other hand Shmuely proved the theorem only for odd-order bases and left the even-order case as an open problem. In this paper we show that the theorem is also true for even-order bases. Precisely speaking we prove that even if there exist a probabilistic polynomial time oracle machine which can solve the problem only for even-order bases still a probabilistic algorithm can be constructed which factors the modulo in polynomial time for more than 98% of RSA-numbers.

• Type: Journal
• Year: 2008
• Where: IJNSEC
• Authors: Kooshiar Azimian, Javad Mohajeri, Mahmoud Salmasizadeh
• Keywords: IJNSEC 2008, Odd-order Bases, Probabilistic Algorithm, Probabilistic Polynomial Time