On the Diffie-Hellman problem over GLn
نویسندگان
چکیده
This paper considers the Diffie-Hellman problem (DHP) over the matrix group GLn over finite fields and shows that for matrices A and exponents k, l satisfying certain conditions called the modulus conditions, the problem can be solved without solving the discrete logarithm problem (DLP) involving only polynomial number of operations in n. A specialization of this result to DHP on Fpm shows that there exists a class of session triples of a DH scheme for which the DHP can be solved in time polynomial in m by operations over Fp without solving the DLP. The private keys of such triples are termed weak. A sample of weak keys is computed and it is observed that their number is not too insignificant to be ignored. Next a specialization of the analysis is carried out for pairing based DH schemes on supersingular elliptic curves and it is shown that for an analogous class of session triples, the DHP can be solved without solving the DLP in polynomial number of operations in the embedding degree. A list of weak parameters of the DH scheme is developed on the basis of this analysis.
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