Summary: A theoretical framework of the bit security of cryptographic primitives/games was first introduced in a pioneering work by D. Micciancio and M. Walter [Lect. Notes Comput. Sci. 10820, 3–28 (2018; Zbl 1423.94090)], and an alternative framework was introduced by the authors [ibid. 13092, 161–188 (2021; Zbl 1514.94139)]. First, we observe that quantitative results in the latter framework are preserved even if adversaries are allowed to output the failure symbol. With this slight modification, we show that the notion of bit security in the latter framework is equivalent to that in the former framework up to constant bits. Also, we demonstrate that several existing notions of advantages can be captured in a unified way. Based on this equivalence, we show that the reduction algorithm of G. Hast [J. Cryptology 17, No. 3, 209–229 (2004; Zbl 1089.94019)] gives a tight reduction of the Goldreich-Levin hard-core predicate to the hardness of one-way functions. These two results resolved open problems that remained.
Furthermore, in the latter framework, we show that all games we need to care about are decision games. Namely, for every search game \(G\), there is the corresponding decision game \(G'\) such that \(G\) has \(\lambda \)-bit security if and only if \(G'\) has \(\lambda \)-bit security. The game \(G'\) consists of the real and the ideal games, where attacks in the ideal game are never approved. Such games often appear in game-hopping security proofs. The result justifies such security proofs because they lose no security. Finally, we provide a distribution replacement theorem. Suppose a game using distribution \(Q\) in a black-box manner is \(\lambda \)-bit secure, and two distributions \(P\) and \(Q\) are computationally \(\lambda \)-bit secure indistinguishable. In that case, the game where \(Q\) is replaced by \(P\) is also \(\lambda \)-bit secure.
For the entire collection see [Zbl 07831382].