Discrete Logarithm Problems with Auxiliary Inputs

Let g be an element of prime order p in an abelian group and let Zp. We show that if g, g , and gd are given for a positive divisor d of p - 1, the secret key can be computed deterministically in O( p/d + d) exponentiations by using O(max{ p/d, d}) storage. If gi (i = 0, 1, 2, . . . , 2d) is given for a positive divisor d of p + 1, can be computed in O( p/d+d) exponentiations by using O(max{ p/d, d}) storage. We also propose space-efficient, but probabilistic algorithms for the same problem, which have the same computational complexities with the deterministic algorithm. As applications of the proposed algorithms, we show that the strong Diffie-Hellman problem and its related problems with public g , . . . , gd have computational complexity up to O( d/ log p) less than the generic algorithm complexity of the discrete logarithm problem when p-1 (resp. p + 1) has a divisor d p1/2 (resp. d p1/3 ). Under the same conditions for d, the algorithm is also applicable to recovering th...

• Type: Journal
• Year: 2010
• Where: JOC
• Authors: Jung Hee Cheon
• Keywords: Algorithm, JOC 2010, Positive Divisor, Secret Key