The multiplicative complexity of 6-variable Boolean functions

The multiplicative complexity of a Boolean function is the minimum number of two-input AND gates that are necessary and sufficient to implement the function over the basis (AND, XOR, NOT). Finding the multiplicative complexity of a given function is computationally intractable, even for functions wi...

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Vydané v:	Cryptography and communications Ročník 11; číslo 1; s. 93 - 107
Hlavní autori:	Çalık, Çağdaş, Sönmez Turan, Meltem, Peralta, René
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	New York Springer US 01.01.2019 Springer Nature B.V
Predmet:	Affine transformations Boolean algebra Boolean functions Boolean Functions and their Applications II Circuits Codes Coding and Information Theory Communications Engineering Complexity Computer Science Data Structures and Information Theory Equivalence Gates Information and Communication Mathematical analysis Mathematics of Computing Networks 06E30 Boolean functions Multiplicative complexity 94A60 Affine equivalence Cryptography Circuit complexity
ISSN:	1936-2447, 1936-2455
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Abstract	The multiplicative complexity of a Boolean function is the minimum number of two-input AND gates that are necessary and sufficient to implement the function over the basis (AND, XOR, NOT). Finding the multiplicative complexity of a given function is computationally intractable, even for functions with small number of inputs. Turan et al. [ 1 ] showed that n -variable Boolean functions can be implemented with at most n − 1 AND gates for n ≤ 5 . A counting argument can be used to show that, for n ≥ 7, there exist n -variable Boolean functions with multiplicative complexity of at least n . In this work, we propose a method to find the multiplicative complexity of Boolean functions by analyzing circuits with a particular number of AND gates and utilizing the affine equivalence of functions. We use this method to study the multiplicative complexity of 6-variable Boolean functions, and calculate the multiplicative complexities of all 150 357 affine equivalence classes. We show that any 6-variable Boolean function can be implemented using at most 6 AND gates. Additionally, we exhibit specific 6-variable Boolean functions which have multiplicative complexity 6.
AbstractList	The multiplicative complexity of a Boolean function is the minimum number of two-input AND gates that are necessary and sufficient to implement the function over the basis (AND, XOR, NOT). Finding the multiplicative complexity of a given function is computationally intractable, even for functions with small number of inputs. Turan et al. [1] showed that n-variable Boolean functions can be implemented with at most n−1 AND gates for n≤5. A counting argument can be used to show that, for n ≥ 7, there exist n-variable Boolean functions with multiplicative complexity of at least n. In this work, we propose a method to find the multiplicative complexity of Boolean functions by analyzing circuits with a particular number of AND gates and utilizing the affine equivalence of functions. We use this method to study the multiplicative complexity of 6-variable Boolean functions, and calculate the multiplicative complexities of all 150 357 affine equivalence classes. We show that any 6-variable Boolean function can be implemented using at most 6 AND gates. Additionally, we exhibit specific 6-variable Boolean functions which have multiplicative complexity 6. The multiplicative complexity of a Boolean function is the minimum number of two-input AND gates that are necessary and sufficient to implement the function over the basis (AND, XOR, NOT). Finding the multiplicative complexity of a given function is computationally intractable, even for functions with small number of inputs. Turan et al. [1] showed that n-variable Boolean functions can be implemented with at most n−1 AND gates for n ≤ 5. A counting argument can be used to show that, for n ≥ 7, there exist n-variable Boolean functions with multiplicative complexity of at least n. In this work, we propose a method to find the multiplicative complexity of Boolean functions by analyzing circuits with a particular number of AND gates and utilizing the affine equivalence of functions. We use this method to study the multiplicative complexity of 6-variable Boolean functions, and calculate the multiplicative complexities of all 150357 affine equivalence classes. We show that any 6-variable Boolean function can be implemented using at most 6 AND gates. Additionally, we exhibit specific 6-variable Boolean functions which have multiplicative complexity 6. The multiplicative complexity of a Boolean function is the minimum number of two-input AND gates that are necessary and sufficient to implement the function over the basis (AND, XOR, NOT). Finding the multiplicative complexity of a given function is computationally intractable, even for functions with small number of inputs. Turan et al. [1] showed that -variable Boolean functions can be implemented with at most -1 AND gates for ≤ 5. A counting argument can be used to show that, for ≥ 7, there exist -variable Boolean functions with multiplicative complexity of at least . In this work, we propose a method to find the multiplicative complexity of Boolean functions by analyzing circuits with a particular number of AND gates and utilizing the affine equivalence of functions. We use this method to study the multiplicative complexity of 6-variable Boolean functions, and calculate the multiplicative complexities of all 150357 affine equivalence classes. We show that any 6-variable Boolean function can be implemented using at most 6 AND gates. Additionally, we exhibit specific 6-variable Boolean functions which have multiplicative complexity 6. The multiplicative complexity of a Boolean function is the minimum number of two-input AND gates that are necessary and sufficient to implement the function over the basis (AND, XOR, NOT). Finding the multiplicative complexity of a given function is computationally intractable, even for functions with small number of inputs. Turan et al. [ 1 ] showed that n -variable Boolean functions can be implemented with at most n − 1 AND gates for n ≤ 5 . A counting argument can be used to show that, for n ≥ 7, there exist n -variable Boolean functions with multiplicative complexity of at least n . In this work, we propose a method to find the multiplicative complexity of Boolean functions by analyzing circuits with a particular number of AND gates and utilizing the affine equivalence of functions. We use this method to study the multiplicative complexity of 6-variable Boolean functions, and calculate the multiplicative complexities of all 150 357 affine equivalence classes. We show that any 6-variable Boolean function can be implemented using at most 6 AND gates. Additionally, we exhibit specific 6-variable Boolean functions which have multiplicative complexity 6. The multiplicative complexity of a Boolean function is the minimum number of two-input AND gates that are necessary and sufficient to implement the function over the basis (AND, XOR, NOT). Finding the multiplicative complexity of a given function is computationally intractable, even for functions with small number of inputs. Turan et al. [1] showed that n-variable Boolean functions can be implemented with at most n-1 AND gates for n ≤ 5. A counting argument can be used to show that, for n ≥ 7, there exist n-variable Boolean functions with multiplicative complexity of at least n. In this work, we propose a method to find the multiplicative complexity of Boolean functions by analyzing circuits with a particular number of AND gates and utilizing the affine equivalence of functions. We use this method to study the multiplicative complexity of 6-variable Boolean functions, and calculate the multiplicative complexities of all 150357 affine equivalence classes. We show that any 6-variable Boolean function can be implemented using at most 6 AND gates. Additionally, we exhibit specific 6-variable Boolean functions which have multiplicative complexity 6.The multiplicative complexity of a Boolean function is the minimum number of two-input AND gates that are necessary and sufficient to implement the function over the basis (AND, XOR, NOT). Finding the multiplicative complexity of a given function is computationally intractable, even for functions with small number of inputs. Turan et al. [1] showed that n-variable Boolean functions can be implemented with at most n-1 AND gates for n ≤ 5. A counting argument can be used to show that, for n ≥ 7, there exist n-variable Boolean functions with multiplicative complexity of at least n. In this work, we propose a method to find the multiplicative complexity of Boolean functions by analyzing circuits with a particular number of AND gates and utilizing the affine equivalence of functions. We use this method to study the multiplicative complexity of 6-variable Boolean functions, and calculate the multiplicative complexities of all 150357 affine equivalence classes. We show that any 6-variable Boolean function can be implemented using at most 6 AND gates. Additionally, we exhibit specific 6-variable Boolean functions which have multiplicative complexity 6.
Author	Çalık, Çağdaş Peralta, René Sönmez Turan, Meltem
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References	Gausdal Find, M.: 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) BoyarJDamgårdIPeraltaRShort non-interactive cryptographic proofsJ Cryptol2000134449472178851510.1007/s001450010011 GoldmanJRotaG-COn the foundations of combinatorial theory iv finite vector spaces and eulerian generating functionsStud. Appl. Math.197049323925826518110.1002/sapm1970493239 Codish, M., Cruz-Filipe, L., Frank, M., Schneider-Kamp, P.: When six gates are not enough. CoRR arXiv:1508.05737 (2015) BraekenABorissovYNikovaSPreneelBClassification of Boolean Functions of 6 Variables or Less with Respect to Some Cryptographic Properties2005HeidelbergBerlin3243341082.94011 Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P.L. (eds.) Boolean Models and Methods in Mathematics, Computer Science and Engineering, chapter 8. Cambridge University Press, Cambridge (2010) SönmezMTPeraltaRThe Multiplicative Complexity of Boolean Functions on Four and Five Variables2015ChamSpringer International Publishing21331382.94167 MaioranaJAA classification of the cosets of the Reed-Muller code $\mathcal {R}$R(1,6)Math. Comput.19915719540341410790270724.94016 CarletClaudeGoubinLouisProuffEmmanuelQuisquaterMichaelRivainMatthieuHigher-Order Masking Schemes for S-BoxesFast Software Encryption2012Berlin, HeidelbergSpringer Berlin Heidelberg36638410.1007/978-3-642-34047-5_21 BerlekampERWelchLRWeight distributions of the cosets of the (32, 6) Reed-Muller codeIEEE Trans. Inf. Theory197218120320739605410.1109/TIT.1972.1054732 Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (leveled) fully homomorphic encryption without bootstrapping. In: Goldwasser, S (ed.) Innovations in Theoretical Computer Science 2012, January 8–10, 2012, pp. 309–325. ACM, Cambridge (2012) NIST Computer Security Division. SLP’s for 6-variable predicates, https://github.com/usnistgov/Circuits/tree/master/slp BoyarJPeraltaRTight bounds for the multiplicative complexity of symmetric functionsTheor. Comput. Sci.20083961–3223246241225710.1016/j.tcs.2008.01.030 KnuthDESubspaces, subsets, and partitionsJ. Comb. Theory, Ser. A197110217818027093310.1016/0097-3165(71)90022-7 Kolesnikov, V., Schneider, T.: Improved garbled circuit: free XOR gates and applications. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewic, 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 (2008) BoyarJPeraltaRPochuevDOn the multiplicative complexity of Boolean functions over the basis (∧, $\oplus $⊕, 1)Theor. Comput. Sci.200023514357176596410.1016/S0304-3975(99)00182-6 Fuller, J.E.: Analysis of affine equivalent boolean functions for cryptography. PhD thesis, Queensland University of Technology (2003) HouX-DAGL (m, 2) acting on R (r, m)/R (s, m)J. Algebra19951713927938131592810.1006/jabr.1995.1043 Claude Carlet (297_CR5) 2012 MT Sönmez (297_CR1) 2015 ER Berlekamp (297_CR11) 1972; 18 X-D Hou (297_CR14) 1995; 171 DE Knuth (297_CR16) 1971; 10 JA Maiorana (297_CR12) 1991; 57 297_CR10 297_CR15 J Boyar (297_CR8) 2008; 396 J Boyar (297_CR9) 2000; 235 J Boyar (297_CR4) 2000; 13 297_CR7 297_CR6 A Braeken (297_CR13) 2005 297_CR18 297_CR3 297_CR2 J Goldman (297_CR17) 1970; 49
References_xml	– reference: BoyarJDamgårdIPeraltaRShort non-interactive cryptographic proofsJ Cryptol2000134449472178851510.1007/s001450010011 – reference: GoldmanJRotaG-COn the foundations of combinatorial theory iv finite vector spaces and eulerian generating functionsStud. Appl. Math.197049323925826518110.1002/sapm1970493239 – reference: Fuller, J.E.: Analysis of affine equivalent boolean functions for cryptography. PhD thesis, Queensland University of Technology (2003) – reference: Codish, M., Cruz-Filipe, L., Frank, M., Schneider-Kamp, P.: When six gates are not enough. CoRR arXiv:1508.05737 (2015) – reference: Carlet, C.: Boolean functions for cryptography and error correcting codes. In: Crama, Y., Hammer, P.L. (eds.) Boolean Models and Methods in Mathematics, Computer Science and Engineering, chapter 8. Cambridge University Press, Cambridge (2010) – reference: BraekenABorissovYNikovaSPreneelBClassification of Boolean Functions of 6 Variables or Less with Respect to Some Cryptographic Properties2005HeidelbergBerlin3243341082.94011 – reference: MaioranaJAA classification of the cosets of the Reed-Muller code $\mathcal {R}$R(1,6)Math. Comput.19915719540341410790270724.94016 – reference: Brakerski, Z., Gentry, C., Vaikuntanathan, V.: (leveled) fully homomorphic encryption without bootstrapping. In: Goldwasser, S (ed.) Innovations in Theoretical Computer Science 2012, January 8–10, 2012, pp. 309–325. ACM, Cambridge (2012) – reference: Gausdal Find, M.: 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) – reference: Kolesnikov, V., Schneider, T.: Improved garbled circuit: free XOR gates and applications. In: Aceto, L., Damgård, I., Goldberg, L.A., Halldórsson, M.M., Ingólfsdóttir, A., Walukiewic, 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 (2008) – reference: NIST Computer Security Division. SLP’s for 6-variable predicates, https://github.com/usnistgov/Circuits/tree/master/slp – reference: CarletClaudeGoubinLouisProuffEmmanuelQuisquaterMichaelRivainMatthieuHigher-Order Masking Schemes for S-BoxesFast Software Encryption2012Berlin, HeidelbergSpringer Berlin Heidelberg36638410.1007/978-3-642-34047-5_21 – reference: KnuthDESubspaces, subsets, and partitionsJ. Comb. Theory, Ser. A197110217818027093310.1016/0097-3165(71)90022-7 – reference: SönmezMTPeraltaRThe Multiplicative Complexity of Boolean Functions on Four and Five Variables2015ChamSpringer International Publishing21331382.94167 – reference: BoyarJPeraltaRTight bounds for the multiplicative complexity of symmetric functionsTheor. Comput. Sci.20083961–3223246241225710.1016/j.tcs.2008.01.030 – reference: HouX-DAGL (m, 2) acting on R (r, m)/R (s, m)J. Algebra19951713927938131592810.1006/jabr.1995.1043 – reference: BerlekampERWelchLRWeight distributions of the cosets of the (32, 6) Reed-Muller codeIEEE Trans. Inf. Theory197218120320739605410.1109/TIT.1972.1054732 – reference: BoyarJPeraltaRPochuevDOn the multiplicative complexity of Boolean functions over the basis (∧, $\oplus $⊕, 1)Theor. Comput. Sci.200023514357176596410.1016/S0304-3975(99)00182-6 – ident: 297_CR7 – volume: 396 start-page: 223 issue: 1–3 year: 2008 ident: 297_CR8 publication-title: Theor. Comput. 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Sci. doi: 10.1016/S0304-3975(99)00182-6 – volume: 13 start-page: 449 issue: 4 year: 2000 ident: 297_CR4 publication-title: J Cryptol doi: 10.1007/s001450010011 – ident: 297_CR15 doi: 10.1017/CBO9780511780448.011 – start-page: 21 volume-title: The Multiplicative Complexity of Boolean Functions on Four and Five Variables year: 2015 ident: 297_CR1 – start-page: 366 volume-title: Fast Software Encryption year: 2012 ident: 297_CR5 doi: 10.1007/978-3-642-34047-5_21 – volume: 171 start-page: 927 issue: 3 year: 1995 ident: 297_CR14 publication-title: J. Algebra doi: 10.1006/jabr.1995.1043 – ident: 297_CR3 – ident: 297_CR10 – ident: 297_CR2 – volume: 18 start-page: 203 issue: 1 year: 1972 ident: 297_CR11 publication-title: IEEE Trans. Inf. Theory doi: 10.1109/TIT.1972.1054732 – volume: 57 start-page: 403 issue: 195 year: 1991 ident: 297_CR12 publication-title: Math. Comput. – volume: 10 start-page: 178 issue: 2 year: 1971 ident: 297_CR16 publication-title: J. Comb. Theory, Ser. 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SubjectTerms	Affine transformations Boolean algebra Boolean functions Boolean Functions and their Applications II Circuits Codes Coding and Information Theory Communications Engineering Complexity Computer Science Data Structures and Information Theory Equivalence Gates Information and Communication Mathematical analysis Mathematics of Computing Networks
Title	The multiplicative complexity of 6-variable Boolean functions
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