# American Institute of Mathematical Sciences

doi: 10.3934/amc.2022013
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## Parameterization of Boolean functions by vectorial functions and associated constructions

 Department of informatics, University of Bergen, Norway and LAGA, University of Paris 8, France

Received  October 2021 Early access March 2022

Fund Project: The research of the author is partly supported by the Trond Mohn Foundation and Norwegian Research Council

Too few general constructions of Boolean functions satisfying all cryptographic criteria are known. We investigate the construction in which the support of $f$ equals the image set of an injective vectorial function $F$, that we call a parameterization of $f$. Every balanced Boolean function can be obtained this way. We study five illustrations of this general construction. The three first correspond to known classes (Maiorana-McFarland, majority functions and balanced functions in odd numbers of variables with optimal algebraic immunity). The two last correspond to new classes:

- the sums of indicators of disjoint graphs of $(k,n-k$)-functions,

- functions parameterized through $(n-1,n)$-functions due to Beelen and Leander.

We study the cryptographic parameters of balanced Boolean functions, according to those of their parameterizations: the algebraic degree of $f$, that we relate to the algebraic degrees of $F$ and of its graph indicator, the nonlinearity of $f$, that we relate by a bound to the nonlinearity of $F$, and the algebraic immunity (AI), whose optimality is related to a natural question in linear algebra, and which may be approached (in two ways) by using the graph indicator of $F$. We revisit each of the five classes for each criterion. The fourth class is very promising, thanks to a lower bound on the nonlinearity by means of the nonlinearity of the chosen $(k,n-k$)-functions. The sub-class of the sums of indicators of affine functions, for which we prove an upper bound and a lower bound on the nonlinearity, seems also interesting. The fifth class includes functions with an optimal algebraic degree, good nonlinearity and good AI.

Citation: Claude Carlet. Parameterization of Boolean functions by vectorial functions and associated constructions. Advances in Mathematics of Communications, doi: 10.3934/amc.2022013
##### References:
 [1] E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, volume 103 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK, 1992.  doi: 10.1017/CBO9781316529836. [2] P. Beelen and G. Leander, A new construction of highly nonlinear S-boxes, Cryptogr. Commun., 4 (2012), 65-77.  doi: 10.1007/s12095-011-0052-4. [3] A. Canteaut, Open problems related to algebraic attacks on stream ciphers, Coding and Cryptography, 3969 (2006), 120-134.  doi: 10.1007/11779360_10. [4] C. Carlet, A larger class of cryptographic Boolean functions via a study of the Maiorana-McFarland construction, Advances in Cryptology—CRYPTO, 2442 (2002), 549-564.  doi: 10.1007/3-540-45708-9_35. [5] C. Carlet, Handling vectorial functions by means of their graph indicators, IEEE Trans. Inform. Theory, 66 (2020), 6324-6339.  doi: 10.1109/TIT.2020.2981524. [6] C. Carlet, Graph indicators of vectorial functions and bounds on the algebraic degree of composite functions, IEEE Trans. Inform. Theory, 66 (2020), 7702-7716.  doi: 10.1109/TIT.2020.3017494. [7] C. Carlet, Boolean Functions for Cryptography and Coding Theory, Monograph in Cambridge University Press 562 pages, 2021. doi: 10.1017/9781108606806. [8] C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156.  doi: 10.1023/A:1008344232130. [9] C. Carlet, D. Dalai, K. Gupta and S. Maitra, Algebraic immunity for cryptographically significant Boolean functions: Analysis and construction, IEEE Trans. Inform. Theory, 52 (2006), 3105-3121.  doi: 10.1109/TIT.2006.876253. [10] C. Carlet and K. Feng, An infinite class of balanced functions with optimum algebraic immunity, good immunity to fast algebraic attacks and good nonlinearity, Advances in cryptology—ASIACRYPT, 5350 (2008), 425-440.  doi: 10.1007/978-3-540-89255-7_26. [11] C. Carlet and P. Méaux, Boolean functions for homomorphic-friendly stream ciphers, Algebra, Codes and Cryptology, 1133 (2019), 166-182.  doi: 10.1007/978-3-030-36237-9_10. [12] C. Carlet and S. Picek, On the practical limits of a generalization of the hidden weight bit function and of another construction of highly nonlinear functions, Preprint, 2022. [13] D. K. Dalai, S. Maitra and S. Sarkar, Basic theory in construction of boolean functions with maximum possible annihilator immunity, Des. Codes Cryptogr., 40 (2006), 41-58.  doi: 10.1007/s10623-005-6300-x. [14] C. Ding, C. Li and Y. Xia, Another generalisation of the binary Reed-Muller codes and its applications, Finite Fields Appl., 53 (2018), 144-174.  doi: 10.1016/j.ffa.2018.06.006. [15] H. Dobbertin, Construction of bent functions and balanced Boolean functions with high nonlinearity, Fast Software Encryption, 1008 (1995), 61-74.  doi: 10.1007/3-540-60590-8_5. [16] I. Dumer and O. Kapralova, Spherically punctured Reed-Muller codes, IEEE Trans. Inform. Theory, 63 (2017), 2773-2780. [17] K. Li, S. Mesnager and L. Qu, Further study of 2-to-1 mappings over $\mathbb{F}_2^n$, IEEE Trans. Inform. Theory, 67 (2021), 3486-3496.  doi: 10.1109/TIT.2021.3057094. [18] F. J. MacWilliams and N. J. Sloane, The Theory of Error-Correcting Codes, Amsterdam-New York-Oxford, 1977. [19] P. Méaux, C. Carlet, A. Journault and F.-X. Standaert, Improved filter permutators for efficient FHE: Better instances and implementations, Proceedings of Indocrypt 2019, Lecture Notes in Computer Science, 11898 (2019), 68-91. [20] P. Méaux, A. Journault, F.-X. Standaert and C. Carlet, Towards stream ciphers for efficient FHE with low-noise ciphertexts, Advances in Cryptology—EUROCRYPT, 9665 (2016), 311-343.  doi: 10.1007/978-3-662-49890-3_13. [21] S. Mesnager, Linear Codes from Functions, Chapter 20 in "A Concise Encyclopedia 1419 Coding Theory" CRC Press/Taylor and Francis Group (Publisher), London, New York, 2021 (94 pages). [22] S. Mesnager and L. Qu, On two-to-one mappings over finite fields, IEEE Trans. Inform. Theory, 65 (2019), 7884-7895.  doi: 10.1109/TIT.2019.2933832. [23] G. Mullen and D. Panario, Handbook of Finite Fields, CRC Press Book, 2013. doi: 10.1201/b15006. [24] Q. Wang, C. Carlet, P. Stănică and C. H. Tan, Cryptographic properties of the hidden weighted bit function, Discrete Appl. Math., 174 (2014), 1-10.  doi: 10.1016/j.dam.2014.01.010.

show all references

##### References:
 [1] E. F. Assmus, Jr. and J. D. Key, Designs and Their Codes, volume 103 of Cambridge Tracts in Mathematics. Cambridge University Press, Cambridge, UK, 1992.  doi: 10.1017/CBO9781316529836. [2] P. Beelen and G. Leander, A new construction of highly nonlinear S-boxes, Cryptogr. Commun., 4 (2012), 65-77.  doi: 10.1007/s12095-011-0052-4. [3] A. Canteaut, Open problems related to algebraic attacks on stream ciphers, Coding and Cryptography, 3969 (2006), 120-134.  doi: 10.1007/11779360_10. [4] C. Carlet, A larger class of cryptographic Boolean functions via a study of the Maiorana-McFarland construction, Advances in Cryptology—CRYPTO, 2442 (2002), 549-564.  doi: 10.1007/3-540-45708-9_35. [5] C. Carlet, Handling vectorial functions by means of their graph indicators, IEEE Trans. Inform. Theory, 66 (2020), 6324-6339.  doi: 10.1109/TIT.2020.2981524. [6] C. Carlet, Graph indicators of vectorial functions and bounds on the algebraic degree of composite functions, IEEE Trans. Inform. Theory, 66 (2020), 7702-7716.  doi: 10.1109/TIT.2020.3017494. [7] C. Carlet, Boolean Functions for Cryptography and Coding Theory, Monograph in Cambridge University Press 562 pages, 2021. doi: 10.1017/9781108606806. [8] C. Carlet, P. Charpin and V. Zinoviev, Codes, bent functions and permutations suitable for DES-like cryptosystems, Des. Codes Cryptogr., 15 (1998), 125-156.  doi: 10.1023/A:1008344232130. [9] C. Carlet, D. Dalai, K. Gupta and S. Maitra, Algebraic immunity for cryptographically significant Boolean functions: Analysis and construction, IEEE Trans. Inform. Theory, 52 (2006), 3105-3121.  doi: 10.1109/TIT.2006.876253. [10] C. Carlet and K. Feng, An infinite class of balanced functions with optimum algebraic immunity, good immunity to fast algebraic attacks and good nonlinearity, Advances in cryptology—ASIACRYPT, 5350 (2008), 425-440.  doi: 10.1007/978-3-540-89255-7_26. [11] C. Carlet and P. Méaux, Boolean functions for homomorphic-friendly stream ciphers, Algebra, Codes and Cryptology, 1133 (2019), 166-182.  doi: 10.1007/978-3-030-36237-9_10. [12] C. Carlet and S. Picek, On the practical limits of a generalization of the hidden weight bit function and of another construction of highly nonlinear functions, Preprint, 2022. [13] D. K. Dalai, S. Maitra and S. Sarkar, Basic theory in construction of boolean functions with maximum possible annihilator immunity, Des. Codes Cryptogr., 40 (2006), 41-58.  doi: 10.1007/s10623-005-6300-x. [14] C. Ding, C. Li and Y. Xia, Another generalisation of the binary Reed-Muller codes and its applications, Finite Fields Appl., 53 (2018), 144-174.  doi: 10.1016/j.ffa.2018.06.006. [15] H. Dobbertin, Construction of bent functions and balanced Boolean functions with high nonlinearity, Fast Software Encryption, 1008 (1995), 61-74.  doi: 10.1007/3-540-60590-8_5. [16] I. Dumer and O. Kapralova, Spherically punctured Reed-Muller codes, IEEE Trans. Inform. Theory, 63 (2017), 2773-2780. [17] K. Li, S. Mesnager and L. Qu, Further study of 2-to-1 mappings over $\mathbb{F}_2^n$, IEEE Trans. Inform. Theory, 67 (2021), 3486-3496.  doi: 10.1109/TIT.2021.3057094. [18] F. J. MacWilliams and N. J. Sloane, The Theory of Error-Correcting Codes, Amsterdam-New York-Oxford, 1977. [19] P. Méaux, C. Carlet, A. Journault and F.-X. Standaert, Improved filter permutators for efficient FHE: Better instances and implementations, Proceedings of Indocrypt 2019, Lecture Notes in Computer Science, 11898 (2019), 68-91. [20] P. Méaux, A. Journault, F.-X. Standaert and C. Carlet, Towards stream ciphers for efficient FHE with low-noise ciphertexts, Advances in Cryptology—EUROCRYPT, 9665 (2016), 311-343.  doi: 10.1007/978-3-662-49890-3_13. [21] S. Mesnager, Linear Codes from Functions, Chapter 20 in "A Concise Encyclopedia 1419 Coding Theory" CRC Press/Taylor and Francis Group (Publisher), London, New York, 2021 (94 pages). [22] S. Mesnager and L. Qu, On two-to-one mappings over finite fields, IEEE Trans. Inform. Theory, 65 (2019), 7884-7895.  doi: 10.1109/TIT.2019.2933832. [23] G. Mullen and D. Panario, Handbook of Finite Fields, CRC Press Book, 2013. doi: 10.1201/b15006. [24] Q. Wang, C. Carlet, P. Stănică and C. H. Tan, Cryptographic properties of the hidden weighted bit function, Discrete Appl. Math., 174 (2014), 1-10.  doi: 10.1016/j.dam.2014.01.010.
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