Abstract
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.
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(a)
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3.
Finally, we show that there exists list-decodable codes with monotone decoders.
In addition, we observe some interesting corollaries to our results.
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Notes
- 1.
A permutation is a length-preserving function that is both injective (i.e., one-to-one) and surjective (i.e., onto).
- 2.
A TCR can be constructed from an SPR by computing a universal hash-function (1-wise independent) on the input before feeding it to the SPR function, namely, masking the inputs with a random key.
- 3.
We write \(a\preceq b\), if for any i, \(i^{th}\) bit of a is 1 implies that the \(i^{th}\) bit of b is 1.
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Miller, D., Scrivener, A., Stern, J., Venkitasubramaniam, M. (2016). On Negation Complexity of Injections, Surjections and Collision-Resistance in Cryptography. In: Dunkelman, O., Sanadhya, S. (eds) Progress in Cryptology – INDOCRYPT 2016. INDOCRYPT 2016. Lecture Notes in Computer Science(), vol 10095. Springer, Cham. https://doi.org/10.1007/978-3-319-49890-4_19
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