Abstract
A ranking function for permutations maps every permutation of length n to a unique integer between 0 and \(n!-1\). For permutations of size that are of interest in cryptographic applications, evaluating such a function requires multiple-precision arithmetic. This work introduces a quasi-optimal ranking technique that allows us to rank a permutation efficiently without needing a multiple-precision arithmetic library. We present experiments that show the computational advantage of our method compared to the standard lexicographic optimal permutation ranking. As an application of our result, we show how this technique improves the signature sizes and the efficiency of PERK digital signature scheme.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
The implementation is available at
References
Aaraj, N., et al.: PERK version 1.0. NIST’s post-quantum cryptography standardization of additional digital signature schemes project (round 1) (2023). https://pqc-perk.org/
Aaraj, N., et al.: PERK version 1.1 (2023). https://pqc-perk.org/resources.html
Bonet, B.: Efficient algorithms to rank and unrank permutations in lexicographic order. In: Workshop on Search in Artificial Intelligence and Robotics - Technical Report (2008)
Chinese Association for Cryptographic Research. National cryptography algorithm design competition (2020). https://www.cacrnet.org.cn/site/content/854.html
Fiat, A., Shamir, A.: How to prove yourself: practical solutions to identification and signature problems. In: Odlyzko, A.M. (ed.) CRYPTO 1986. LNCS, vol. 263, pp. 186–194. Springer, Heidelberg (1987). https://doi.org/10.1007/3-540-47721-7_12
Ishai, Y., Kushilevitz, E., Ostrovsky, R., Sahai, A.: Zero-knowledge from secure multiparty computation. In: Proceedings of the 39th annual ACM symposium on Theory of computing (STOC) (2007)
Kannwischer, M.J., Krausz, M., Petri, R., Yang, S.-J.: pqm4: benchmarking NIST additional post-quantum signature schemes on microcontrollers. Cryptology ePrint Archive, Paper 2024/112 (2024). https://eprint.iacr.org/2024/112
Lehmer, D.H.: Teaching combinatorial tricks to a computer. Combin. Anal. 179–193 (1960)
Leon, J.: Computing automorphism groups of error-correcting codes. IEEE Trans. Inf. Theory 28(3), 496–511 (1982)
Mytkowicz, T., Diwan, A., Hauswirth, M., Sweeney, P.F.: Producing wrong data without doing anything obviously wrong! ACM Sigplan Not. 44(3), 265–276 (2009)
Myrvold, W., Ruskey, F.: Ranking and unranking permutations in linear time. Inf. Process. Lett. 79(6), 281–284 (2001)
NIST. Post-quantum cryptography standardization (2017). https://csrc.nist.gov/projects/post-quantum-cryptography
NIST. Post-quantum cryptography: Digital signature schemes (2023). https://csrc.nist.gov/Projects/pqc-dig-sig/round-1-additional-signatures
The GNU Project. GMP: The GNU Multiple Precision Arithmetic Library (2023). https://gmplib.org/. [version 6.2.1]
Russell, S., Norvig, P.: Artificial Intelligence: A Modern Approach, 3rd edn. Prentice Hall Press, USA (2009)
Shamir, A.: An efficient identification scheme based on permuted kernels (extended abstract). In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 606–609. Springer, New York (1990). https://doi.org/10.1007/0-387-34805-0_54
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2024 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Bettaieb, S., Budroni, A., Palumbi, M., Filho, D.L.G. (2024). Quasi-optimal Permutation Ranking and Applications to PERK. In: Vaudenay, S., Petit, C. (eds) Progress in Cryptology - AFRICACRYPT 2024. AFRICACRYPT 2024. Lecture Notes in Computer Science, vol 14861. Springer, Cham. https://doi.org/10.1007/978-3-031-64381-1_3
Download citation
DOI: https://doi.org/10.1007/978-3-031-64381-1_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-64380-4
Online ISBN: 978-3-031-64381-1
eBook Packages: Computer ScienceComputer Science (R0)