Boneh-Boyen Signatures and the Strong Diffie-Hellman Problem

Abstract

Abstract. The Boneh-Boyen signature scheme is a pairing based short signature scheme which is provably secure in the standard model under the q-Strong Diffie-Hellman assumption. In this paper, we prove the converse of this statement, and show that forging Boneh-Boyen signatures is actually equivalent to solving the q-Strong Diffie-Hellman problem. Using this equivalence, we exhibit an algorithm which, on the vast majority of pairing-friendly curves, recovers Boneh-Boyen private keys in O(p 2 5 +ε ) time, using O(p 1 5 +ε ) signature queries. We present implementation results comparing the performance of our algorithm and traditional discrete logarithm algorithms such as Pollard's lambda algorithm and Pollard's rho algorithm. We also discuss some possible countermeasures and strategies for mitigating the impact of these findings.

Solving DLP with Auxiliary Input over an Elliptic Curve Used in TinyTate Library

“…Estimated Time for 128-bit r Estimated Time Jao, Yoshida [11] 16384 Days Izu et al [10] 1195 Days This paper 7 Days…”

“…Also note that Izu et al's result was implemented over a finite field with characteristics 3). Here, solving Size of r Required Time Jao, Yoshida [11] 60 bit 3 hours Izu et al [10] 83 bit 14 hours This paper 128 bit 45 hours DLP on the elliptic curve is regarded to be infeasible (since the order is 128-bit).…”

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“…The first realization of this importance was done by Brown and Gallant [9] and Cheon [10,11]. Brown and Gallant presented an algorithm to compute α for given g, g α , g or p + 1 has a small divisor d. Jao and Yoshida [14] gave an algorithm to forge the Boneh-Boyen signatures using the Cheon's algorithm.…”