Design Validations for Discrete Logarithm Based Signature Schemes

A number of signature schemes and standards have been recently designed, based on the Discrete Logarithm problem. In this paper we conduct design validation of such schemes while trying to minimize the use of ideal hash functions. We consider several Discrete Logarithm (DSA-like) signatures abstracted as generic schemes. We show that the following holds: “if the schemes can be broken by an existential forgery using an adaptively chosen-message attack then either the discrete logarithm problem can be solved, or some hash function can be distinguished from an ideal one, or multicollisions can be found.” Thus, for these signature schemes, either they are equivalent to the discrete logarithm problem or there is an attack that takes advantage of properties which are not desired (or expected) in strong practical hash functions (SHA-1 or whichever high quality cryptographic hash function is used). What is interesting is that the schemes we discuss include KCDSA and slight variations of DSA. Further, since our schemes coincide with (or are extremely close to) their standard counterparts they benefit from their desired properties: efficiency of computation/space, employment of certain mathematical operations and wide applicability to various algebraic structures. We feel that adding variants with strong validation of security is important to this family of signature schemes since, as we have experienced in the recent past, lack of such validation has led to attacks on standard schemes, years after their introduction. In addition, schemes with formal validation which is made public, may ease global standardization since they neutralize much of the suspicions regarding potential knowledge gaps and unfair advantages gained by the scheme designer’s country (e.g. the NSA being the designers of DSA).