Boolean functions with multiplicative complexity 3 and 4

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 ), respe...

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Published in:	Cryptography and communications Vol. 12; no. 5; pp. 935 - 946
Main Authors:	Çalık, Çağdaş, Turan, Meltem Sönmez, Peralta, René
Format:	Journal Article
Language:	English
Published:	New York Springer US 01.09.2020 Springer Nature B.V
Subjects:	Affine transformations Boolean Boolean algebra Boolean functions Circuits Codes Coding and Information Theory Communications Engineering Complexity Computer Science Data Structures and Information Theory Equivalence Gates (circuits) Information and Communication Linearity Lower bounds Mathematics of Computing Networks 06E30 Boolean functions 94A60 Affine equivalence Multiplicative complexity
ISSN:	1936-2447, 1936-2455
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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 d i m ( 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 ⌈ d i m ( 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.
AbstractList	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 d i m ( 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 ⌈ d i m ( 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. 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. 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 Fischer and Peralta (2002), and Find et al. (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.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 Fischer and Peralta (2002), and Find et al. (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. 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 Fischer and Peralta (2002), and Find et al. (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. 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 Fischer and Peralta (2002), and Find et al. (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 ( ) of a Boolean function in relation to its linearity dimension, and provide a new lower bound suggesting that the multiplicative complexity of is at least [ ( )/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 -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.
Author	Çalık, Çağdaş Peralta, René Turan, Meltem Sönmez
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References	Schnorr, C-P: The multiplicative complexity of Boolean functions. In: AAECC, pp 45–58 (1988) 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) 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) FindMGSmith-ToneDTuranMSThe number of Boolean functions with multiplicative complexity 2IJICoT201744222236370869410.1504/IJICOT.2017.086890 ÇalıkÇTuranMSPeraltaRThe multiplicative complexity of 6-variable boolean functionsCryptogr. Commun.201911193107389582510.1007/s12095-018-0297-2 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) HouX-DAGL(m,2) acting on R(r, m)/R(s,m)J. Algebra19951713927938131592810.1006/jabr.1995.1043 MirwaldRSchnorrC-PThe multiplicative complexity of quadratic Boolean formsTheor. Comput. Sci.19921022307328117473710.1016/0304-3975(92)90235-8 Preneel, B.: Analysis and design of cryptographic hash functions. PhD thesis, Katholieke Universiteit Leuven (1993) BoyarJPeraltaRPochuevDOn the multiplicative complexity of Boolean functions over the basis (∧, ⊕, 1)Theor. Comput. Sci.200023514357176596410.1016/S0304-3975(99)00182-6 Uyan, E: Analysis of Boolean functions with respect to Walsh Spectrum. PhD thesis, Middle East Technical University (2013) Fuller, JE: Analysis of affine equivalent boolean functions for cryptography. PhD thesis, Queensland University of Technology (2003) NIST Computer Security Division. Circuit Complexity Project Repository, https://github.com/usnistgov/Circuits 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) BerlekampERWelchLRWeight distributions of the cosets of the (32, 6) Reed-Muller codeIEEE Trans. Inf. Theory197218120320739605410.1109/TIT.1972.1054732 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) TuranMSPeraltaRThe Multiplicative Complexity of Boolean Functions on Four and Five Variables2015ChamSpringer International Publishing21331382.94167 Fischer, M. J., Peralta, R.: Counting Predicates of Conjunctive Complexity One. Yale Technical Report 1222 (2002) BoyarJDamgårdIPeraltaRShort non-interactive cryptographic proofsJ. Cryptol.2000134449472178851510.1007/s001450010011 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) BrandãoLTANÇalıkÇTuranMSPeraltaRUpper bounds on the multiplicative complexity of symmetric boolean functionsCryptogr. Commun.201911613391362403317810.1007/s12095-019-00377-3 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) MaioranaJAA classification of the cosets of the Reed-Muller code R(1,6)Math. Comput.19915719540341410790270724.94016 Ç Çalık (445_CR8) 2019; 11 LTAN Brandão (445_CR7) 2019; 11 X-D Hou (445_CR14) 1995; 171 MG Find (445_CR11) 2017; 4 JA Maiorana (445_CR16) 1991; 57 445_CR6 445_CR5 445_CR18 445_CR19 445_CR1 ER Berlekamp (445_CR2) 1972; 18 445_CR15 J Boyar (445_CR3) 2000; 13 445_CR10 445_CR21 MS Turan (445_CR22) 2015 445_CR12 445_CR23 445_CR9 445_CR13 R Mirwald (445_CR17) 1992; 102 445_CR20 J Boyar (445_CR4) 2000; 235
References_xml	– reference: 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) – reference: Preneel, B.: Analysis and design of cryptographic hash functions. PhD thesis, Katholieke Universiteit Leuven (1993) – reference: Schnorr, C-P: The multiplicative complexity of Boolean functions. In: AAECC, pp 45–58 (1988) – reference: TuranMSPeraltaRThe Multiplicative Complexity of Boolean Functions on Four and Five Variables2015ChamSpringer International Publishing21331382.94167 – reference: 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) – reference: MirwaldRSchnorrC-PThe multiplicative complexity of quadratic Boolean formsTheor. Comput. Sci.19921022307328117473710.1016/0304-3975(92)90235-8 – reference: 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) – reference: ÇalıkÇTuranMSPeraltaRThe multiplicative complexity of 6-variable boolean functionsCryptogr. Commun.201911193107389582510.1007/s12095-018-0297-2 – reference: BoyarJDamgårdIPeraltaRShort non-interactive cryptographic proofsJ. Cryptol.2000134449472178851510.1007/s001450010011 – reference: 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) – reference: FindMGSmith-ToneDTuranMSThe number of Boolean functions with multiplicative complexity 2IJICoT201744222236370869410.1504/IJICOT.2017.086890 – reference: Fuller, JE: Analysis of affine equivalent boolean functions for cryptography. PhD thesis, Queensland University of Technology (2003) – reference: 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) – reference: BrandãoLTANÇalıkÇTuranMSPeraltaRUpper bounds on the multiplicative complexity of symmetric boolean functionsCryptogr. Commun.201911613391362403317810.1007/s12095-019-00377-3 – reference: Uyan, E: Analysis of Boolean functions with respect to Walsh Spectrum. PhD thesis, Middle East Technical University (2013) – reference: Fischer, M. J., Peralta, R.: Counting Predicates of Conjunctive Complexity One. Yale Technical Report 1222 (2002) – reference: 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) – reference: MaioranaJAA classification of the cosets of the Reed-Muller code R(1,6)Math. Comput.19915719540341410790270724.94016 – reference: 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) – reference: NIST Computer Security Division. Circuit Complexity Project Repository, https://github.com/usnistgov/Circuits/ – reference: BerlekampERWelchLRWeight distributions of the cosets of the (32, 6) Reed-Muller codeIEEE Trans. Inf. Theory197218120320739605410.1109/TIT.1972.1054732 – reference: HouX-DAGL(m,2) acting on R(r, m)/R(s,m)J. Algebra19951713927938131592810.1006/jabr.1995.1043 – reference: BoyarJPeraltaRPochuevDOn the multiplicative complexity of Boolean functions over the basis (∧, ⊕, 1)Theor. Comput. Sci.200023514357176596410.1016/S0304-3975(99)00182-6 – volume: 102 start-page: 307 issue: 2 year: 1992 ident: 445_CR17 publication-title: Theor. Comput. Sci. doi: 10.1016/0304-3975(92)90235-8 – volume: 4 start-page: 222 issue: 4 year: 2017 ident: 445_CR11 publication-title: IJICoT doi: 10.1504/IJICOT.2017.086890 – start-page: 21 volume-title: The Multiplicative Complexity of Boolean Functions on Four and Five Variables year: 2015 ident: 445_CR22 – volume: 171 start-page: 927 issue: 3 year: 1995 ident: 445_CR14 publication-title: J. 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Snippet	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)....
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SubjectTerms	Affine transformations Boolean Boolean algebra Boolean functions Circuits Codes Coding and Information Theory Communications Engineering Complexity Computer Science Data Structures and Information Theory Equivalence Gates (circuits) Information and Communication Linearity Lower bounds Mathematics of Computing Networks
Title	Boolean functions with multiplicative complexity 3 and 4
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