Computationally sound analysis of protocols using bilinear pairings

In this paper, we introduce a symbolic model to analyse protocols that use a bilinear pairing between two cyclic groups. This model consists in an extension of the Abadi-Rogaway logic and we prove that the logic is still computationally sound: symbolic indistinguishability implies computational indistinguishability provided that the Bilinear Decisional DiffieHellman assumption holds and that the encryption scheme is IND-CPA secure. We illustrate our results on classical protocols using bilinear pairing like Joux tripartite Diffie-Hellman protocol or the TAK-2 and TAK-3 protocols. We also investigate the security of a newly designed variant of the Burmester-Desmedt protocol using bilinear pairings. More precisely, we show for each of these protocols that the generated key is indistinguishable from a random element.