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.
-
(a)
-
3.
Finally, we show that there exists list-decodable codes with monotone decoders.
In addition, we observe some interesting corollaries to our results.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
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.
References
Amano, K., Maruoka, A.: A superpolynomial lower bound for a circuit computing the clique function with at most (1/6) log log n negation gates. SIAM J. Comput. 35(1), 201–216 (2005)
Beals, R., Nishino, T., Tanaka, K.: More on the complexity of negation-limited circuits. In: Proceedings of the Twenty-Seventh Annual ACM Symposium on Theory of Computing, Las Vegas, Nevada, USA, 29 May–1 June 1995, pp. 585–595 (1995)
Beals, R., Nishino, T., Tanaka, K.: On the complexity of negation-limited Boolean networks. SIAM J. Comput. 27(5), 1334–1347 (1998)
Blais, E., Canonne, C.L., Oliveira, I.C., Servedio, R.A., Tan, L.: Learning circuits with few negations. CoRR abs/1410.8420 (2014)
Blum, A., Burch, C., Langford, J.: On learning monotone Boolean functions. In: 39th Annual Symposium on Foundations of Computer Science, FOCS 1998, Palo Alto, California, USA, 8–11 November 1998, pp. 408–415 (1998)
Buresh-Oppenheim, J., Kabanets, V., Santhanam, R.: Uniform hardness amplification in NP via monotone codes. Electron. Colloquium Comput. Complex. (ECCC) 13(154) (2006)
Fischer, M.J.: The complexity of negation-limited networks — a brief survey. In: Brakhage, H. (ed.) GI-Fachtagung 1975. LNCS, vol. 33, pp. 71–82. Springer, Heidelberg (1975). doi:10.1007/3-540-07407-4_9
Goldreich, O., Izsak, R.: Monotone circuits: one-way functions versus pseudorandom generators. Theory Comput. 8(1), 231–238 (2012)
Guo, S., Malkin, T., Oliveira, I.C., Rosen, A.: The power of negations in cryptography. In: Dodis, Y., Nielsen, J.B. (eds.) TCC 2015. LNCS, vol. 9014, pp. 36–65. Springer, Heidelberg (2015). doi:10.1007/978-3-662-46494-6_3
Karchmer, M., Wigderson, A.: Monotone circuits for connectivity require super-logarithmic depth. In: Proceedings of the 20th Annual ACM Symposium on Theory of Computing, Chicago, Illinois, USA, 2–4 May 1988, pp. 539–550 (1988)
Knuth, D.E.: Efficient balanced codes. IEEE Trans. Inf. Theory 32(1), 51–53 (1986)
Markov, A.A.: On the inversion complexity of a system of functions. J. ACM 5(4), 331–334 (1958)
Morizumi, H.: Limiting negations in non-deterministic circuits. Theor. Comput. Sci. 410(38–40), 3988–3994 (2009)
Naor, M., Yung, M.: Universal one-way hash functions and their cryptographic applications. In: Proceedings of the 21st Annual ACM Symposium on Theory of Computing, Seattle, Washigton, USA, 14–17 May 1989, pp. 33–43 (1989)
Santha, M., Wilson, C.B.: Limiting negations in constant depth circuits. SIAM J. Comput. 22(2), 294–302 (1993)
Sung, S.C., Tanaka, K.: Limiting negations in bounded-depth circuits: an extension of Markov’s theorem. Inf. Process. Lett. 90(1), 15–20 (2004)
Tardos, É.: The gap between monotone and non-monotone circuit complexity is exponential. Combinatorica 8(1), 141–142 (1988)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing AG
About this paper
Cite this paper
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
Download citation
DOI: https://doi.org/10.1007/978-3-319-49890-4_19
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-49889-8
Online ISBN: 978-3-319-49890-4
eBook Packages: Computer ScienceComputer Science (R0)