A Practical and Tightly Secure Signature Scheme Without Hash Function

In 1999, two signature schemes based on the flexible RSA problem (a.k.a. strong RSA problem) were independently introduced: the Gennaro-Halevi-Rabin (GHR) signature scheme and the Cramer-Shoup (CS) signature scheme. Remarkably, these schemes meet the highest security notion in the standard model. They however differ in their implementation. The CS scheme and its subsequent variants and extensions proposed so far feature a loose security reduction, which, in turn, implies larger security parameters. The security of the GHR scheme and of its twinning-based variant are shown to be tightly based on the flexible RSA problem but additionally (i) either assumes the existence of division-intractable hash functions, or (ii) requires an injective mapping into the prime numbers in both the signing and verification algorithms. In this paper, we revisit the GHR signature scheme and completely remove the extra assumption made on the hash functions without relying on injective prime mappings. As a result, we obtain a practical signature scheme (and an on-line/off-line variant thereof) whose security is solely and tightly related to the strong RSA assumption.