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

This paper develops a cryptanalysis of the pairing based Diffie Hellman (DH) key exchange schemes which have found important applications as in the tripartite exchange scheme proposed in [1]. The analysis of weak keys of the standard DH scheme proposed in [2] is applied to show existence of weak sessions for tripartite schemes over supersingular curves. It is shown that for such sessions the associated Bilinear Diffie Hellman Problem (BDHP) is solvable in polynomial time, without computing the private keys i.e. without solving the discrete logarithms. Similar applications of the analysis to Decisional Diffie Hellman Problem (DDHP)and the Identity Based DH scheme (IBS) are also developed. The tripartite key exchange scheme is analyzed in detail and it is shown that the number of weak keys increases in this scheme as compared to the standard two party DH scheme. It is shown that the random choice of private keys by the users independent of each other’s knowledge is insecure in these schemes. Algorithms are suggested for checking weakness of private keys based on an order of selection. A modified tripartite key exchange scheme is presented in which detection of weak keys is incorporated.