Improving the exact security of digital signature schemes

We put forward a new method of constructing Fiat-Shamir-like signature schemes that yields better "exact security" than the original Fiat-Shamir method. (We also point out, however, that such tight security does not make our modified schemes always preferable to the original ones, Indeed, there exist particularly efficient Fiat-Shamir-like schemes that, though only enjoying "loose security," by using longer keys may provably provide more security at a lower computational cost than their "tight-security" counterparts).