Salient in many cryptosystems, the exponent-inversion technique began without randomization in the random oracle model (SCIS '03, PKC '04), evolved into the Boneh-Boyen short signature scheme (JoC '08) and exerted a wide influence. Seen as a notable case, Gentry's (EuroCrypt '06) identity-based encryption (IBE) applies exponent inversion on a randomized base in its identity-based trapdoors. Making use of the non-static q-strong Diffie-Hellman assumption, Boneh-Boyen signatures are shown to be unforgeable against q-chosen-message attacks, while a variant q-type decisional assumption is used to establish the security of Gentry-IBE. Challenges remain in proving their security under weaker static assumptions.

Supported by the dual form/system framework (Crypto '09, AsiaCrypt '12), we propose dual form exponent-inversion Boneh-Boyen signatures and Gentry-IBE, with security proven under the symmetric external Diffie-Hellman (SXDH) assumption. Starting from our signature scheme, we extend it into P-signatures (TCC '08), resulting in the first anonymous credential scheme from the SXDH assumption, serving as a competitive alternative to the static-assumption construction of Abe et al. (JoC '16). Moreover, from our Gentry-IBE variant, we propose an accountable-authority IBE scheme also from SXDH, surpassing the fully secure Sahai-Seyalioglu scheme (PKC '11) in efficiency and the generic Kiayias-Tang transform (ESORICS '15) in security. Collectively, we present a suite of results under static assumptions.