In this paper, we present three digital signature schemes with tight security reductions in the random oracle model. Our first signature scheme is a particularly efficient version of the short exponent discrete log-based scheme of Girault et al. (J Cryptol 19(4):463–487, 2006). Our scheme has a tight reduction to the decisional short discrete logarithm problem, while still maintaining the non-tight reduction to the computational version of the problem upon which the original scheme of Girault et al. is based. The second signature scheme we construct is a modification of the scheme of Lyubashevsky (Advances in Cryptology—ASIACRYPT 2009, vol 5912 of Lecture Notes in Computer Science, pp 598–616, Tokyo, Japan, December 6–10, 2009. Springer, Berlin, 2009) that is based on the worst-case hardness of the shortest vector problem in ideal lattices. And the third scheme is a very simple signature scheme that is based directly on the hardness of the subset sum problem. We also present a general transformation that converts what we term lossy identification schemes into signature schemes with tight security reductions. We believe that this greatly simplifies the task of constructing and proving the security of such signature schemes.