Goldreich and Izsak (Theory of Computing, 2012) initiated the research on understanding the role of negations in circuits implementing cryptographic primitives, notably, considering one-way functions and pseudo-random generators. More recently, Guo, Malkin, Oliveira and Rosen (TCC, 2015) determined tight bounds on the minimum number of negations gates (i.e., negation complexity) of a wide variety of cryptographic primitives including pseudo-random functions, error-correcting codes, hardcore-predicates and randomness extractors. We continue this line of work to establish the following results: 1. First, we determine tight lower bounds on the negation complexity of collision-resistant and target collision-resistant hash-function families. 2. Next, we examine the role of injectivity and surjectivity on the negation complexity of one-way functions. Here we show that, (a) Assuming the existence of one-way injections, there exists a monotone one-way injection. Furthermore, we complement our result by showing that, even in the worst-case, there cannot exist a monotone one-way injection with constant stretch. (b) Assuming the existence of one-way permutations, there exists a monotone one-way surjection. 3. Finally, we show that there exists list-decodable codes with monotone decoders. In addition, we observe some interesting corollaries to our results.
CITATION STYLE
Miller, D., Scrivener, A., Stern, J., & Venkitasubramaniam, M. (2016). On negation complexity of injections, surjections and collision-resistance in cryptography. In Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) (Vol. 10095 LNCS, pp. 345–363). Springer Verlag. https://doi.org/10.1007/978-3-319-49890-4_19
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