Weak keys of the Diffie Hellman key exchange II : Pairing based schemes on elliptic curves

This paper investigates the Diffie-Hellman key exchange scheme over the group F∗ p of nonzero elements of finite fields and shows that there exist exponents k, l satisfying certain conditions called the modulus conditions, for which the Diffie Hellman Problem (DHP) can be solved in polynomial number of operations in m without solving the discrete logarithm problem (DLP). These special private keys of the scheme are termed weak and depend also on the generator a of the cyclic group. More generally the triples (a, k, l) with generator a and one of private keys k, l weak, are called weak triples. A sample of weak keys is computed and it is observed that their number may not be insignificant to be ignored in general. Next, an extension of the analysis and weak triples is carried out for the Diffie Hellman scheme over the matrix group GLn 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 matrix size n. A revised Diffie Hellman assumption is stated, taking into account the above exceptions.