Abstract
Multiplicative complexity (MC) is defined as the minimum number of AND gates required to implement a function with a circuit over the basis (AND, XOR, NOT). Boolean functions with MC 1 and 2 have been characterized in Fisher and Peralta (2002), and Find et al. (IJICoT 4(4), 222–236, 2017), respectively. In this work, we identify the affine equivalence classes for functions with MC 3 and 4. In order to achieve this, we utilize the notion of the dimension dim(f) of a Boolean function in relation to its linearity dimension, and provide a new lower bound suggesting that the multiplicative complexity of f is at least ⌈dim(f)/2⌉. For MC 3, this implies that there are no equivalence classes other than those 24 identified in Çalık et al. (2018). Using the techniques from Çalık et al. and the new relation between the dimension and MC, we identify all 1277 equivalence classes having MC 4. We also provide a closed formula for the number of n-variable functions with MC 3 and 4. These results allow us to construct AND-optimal circuits for Boolean functions that have MC 4 or less, independent of the number of variables they are defined on.
Similar content being viewed by others
Notes
We abuse notation here, identifying a node with the function it computes.
References
Albrecht, MR, Rechberger, C, Schneider, T, Tiessen, T, Zohner, M: Ciphers for MPC and FHE. In: Oswald, E, Fischlin, M (eds.) Advances in Cryptology—EUROCRYPT 2015—34th Annual International Conference on the Theory and Applications of Cryptographic Techniques, Sofia, Bulgaria, April 26–30, 2015, Proceedings, Part I, volume 9056 of Lecture Notes in Computer Science, pp 430–454. Springer, Berlin (2015)
Berlekamp, ER, Welch, LR: Weight distributions of the cosets of the (32, 6) Reed-Muller code. IEEE Trans. Inf. Theory 18(1), 203–207 (1972)
Boyar, J, Damgård, I, Peralta, R: Short non-interactive cryptographic proofs. J. Cryptol. 13(4), 449–472 (2000)
Boyar, J, Peralta, R, Pochuev, D: On the multiplicative complexity of Boolean functions over the basis (∧, ⊕, 1). Theor. Comput. Sci. 235(1), 43–57 (2000)
Braeken, A, Borissov, YL, Nikova, S, Preneel, B: Classification of Boolean functions of 6 variables or less with respect to some cryptographic properties. In: Caires, L, Italiano, GF, Monteiro, L, Palamidessi, C, Yung, M (eds.) ICALP, volume 3580 of Lecture Notes in Computer Science, pp 324–334. Springer, Berlin (2005)
Brakerski, Z, Gentry, C, Vaikuntanathan, V: (Leveled) fully homomorphic encryption without bootstrapping. In: Goldwasser, S (ed.) Innovations in Theoretical Computer Science, January 8–10, 2012, p 2012, Cambridge (2012)
Brandão, L.T. A. N., Çalık, Ç, Turan, MS, Peralta, R: Upper bounds on the multiplicative complexity of symmetric boolean functions. Cryptogr. Commun. 11(6), 1339–1362 (2019)
Çalık, Ç, Turan, MS, Peralta, R: The multiplicative complexity of 6-variable boolean functions. Cryptogr. Commun. 11(1), 93–107 (2019)
Dobraunig, C, Eichlseder, M, Grassi, L, Lallemand, V, Leander, G, List, E, Mendel, F, Rechberger, C: Rasta: a cipher with low ANDdepth and few ANDs per bit. In: CRYPTO (1), volume 10991 of Lecture Notes in Computer Science, pp 662–692. Springer, Berlin (2018)
Find, MG: On the complexity of computing two nonlinearity measures. In: Computer Science - Theory and Applications - 9th International Computer Science Symposium in Russia, CSR 2014, Moscow, Russia, June 7–11, 2014. Proceedings, pp 167–175 (2014)
Find, MG, Smith-Tone, D, Turan, MS: The number of Boolean functions with multiplicative complexity 2. IJICoT 4(4), 222–236 (2017)
Fischer, M. J., Peralta, R.: Counting Predicates of Conjunctive Complexity One. Yale Technical Report 1222 (2002)
Fuller, JE: Analysis of affine equivalent boolean functions for cryptography. PhD thesis, Queensland University of Technology (2003)
Hou, X-D: AGL(m,2) acting on R(r, m)/R(s,m). J. Algebra 171(3), 927–938 (1995)
Kolesnikov, V, Schneider, T: Improved garbled circuit: free XOR gates and applications. In: Aceto, L, Damgård, I, Goldberg, L.A., Halldórsson, MM, Ingólfsdóttir, A, Walukiewicz, I (eds.) Automata, Languages and Programming, 35th International Colloquium, ICALP 2008, Reykjavik, Iceland, July 7–11, 2008, Proceedings, Part II - Track B: Logic, Semantics, and Theory of Programming & Track C: Security and Cryptography Foundations, volume 5126 of Lecture Notes in Computer Science, pp 486–498. Springer, Berlin (2008)
Maiorana, J.A.: A classification of the cosets of the Reed-Muller code R(1,6). Math. Comput. 57(195), 403–414 (1991)
Mirwald, R, Schnorr, C-P: The multiplicative complexity of quadratic Boolean forms. Theor. Comput. Sci. 102(2), 307–328 (1992)
NIST Computer Security Division. Circuit Complexity Project Repository, https://github.com/usnistgov/Circuits/
Nyberg, K: On the construction of highly nonlinear permutations. In: Rueppel, RA (ed.) Advances in Cryptology - EUROCRYPT ’92, Workshop on the Theory and Application of of Cryptographic Techniques, Balatonfüred, Hungary, May 24–28, 1992, Proceedings, volume 658 of Lecture Notes in Computer Science, pp 92–98. Springer, Berlin (1992)
Preneel, B.: Analysis and design of cryptographic hash functions. PhD thesis, Katholieke Universiteit Leuven (1993)
Schnorr, C-P: The multiplicative complexity of Boolean functions. In: AAECC, pp 45–58 (1988)
Turan, MS, Peralta, R: The Multiplicative Complexity of Boolean Functions on Four and Five Variables, pp 21–33. Springer International Publishing, Cham (2015)
Uyan, E: Analysis of Boolean functions with respect to Walsh Spectrum. PhD thesis, Middle East Technical University (2013)
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Çalık, Ç., Turan, M.S. & Peralta, R. Boolean functions with multiplicative complexity 3 and 4. Cryptogr. Commun. 12, 935–946 (2020). https://doi.org/10.1007/s12095-020-00445-z
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s12095-020-00445-z