On primary functions which are minimally close to linear functions. (Russian. English summary) Zbl 1475.94169
Summary: The investigation of aspects of closeness to linear functions for functions from \((\mathbb{Z}/(p))^n\) to \((\mathbb{Z}/(p))^m\) (\(p\) is prime number). New criteria of minimal closeness to linear functions are found. This property of a function is proved to be inherited for its homomorphic images. As a generalization of an analogous statement for Boolean functions it is proved that if \(p=2\) or 3 then a class of functions which are minimally close to linear ones coincides with the class of bent-functions (if bent-functions do exist).
Keywords:
closeness of functions; absolutely nonhomomorphic functions; minimal functions; bent-functionsReferences:
| [1] | Golomb S. W., “On the classification of Boolean functions”, IRE Trans. Circuit Theory, 1:6 (1959), 10-27 |
| [2] | Rothaus O. S., “On “bent” functions”, J. Comb. Theory Ser. A, 20:3 (1976), 300-305 · Zbl 0336.12012 · doi:10.1016/0097-3165(76)90024-8 |
| [3] | Ambrosimov A. S., “Svoistva bent-funktsii \(q\)-znachnoi logiki nad konechnymi polyami”, Diskretnaya matematika, 6:3 (1994), 50-60 · Zbl 0816.03010 |
| [4] | Logachëv O. A., Salnikov A. A., Yaschenko V. V., “Bent-funktsii na konechnoi abelevoi gruppe”, Diskretnaya matematika, 9:4 (1997), 3-20 · Zbl 0982.94012 |
| [5] | Solodovnikov V. I., “Bent-funktsii iz konechnoi abelevoi gruppy v konechnuyu abelevu gruppu”, Diskretnaya matematika, 14:1 (2002), 99-113 · Zbl 1047.94011 |
| [6] | Kuzmin A. S., Nechaev A. A., Shishkin V. A., “Bent- i giperbent-funktsii nad konechnym polem”, Trudy po diskretnoi matematike, 10, Fizmatlit, M., 2007, 97-122 |
| [7] | Kuzmin A. S., Nechaev A. A., Shishkin V. A., “Parametry (giper-) bent-funktsii nad polem iz \(2^l\) elementov”, Trudy po diskretnoi matematike, 11, no. 1, Fizmatlit, M., 2008, 47-59 |
| [8] | Kumar P. V., Scholts R. A., Welch L. R., “Generalized bent functions and their properties”, J. Comb. Theory Ser. A, 40:1 (1985), 90-107 · Zbl 0585.94016 · doi:10.1016/0097-3165(85)90049-4 |
| [9] | Nyberg K., “Perfect nonlinear \(S\)-boxes”, Adv. in Cryptology - EURO-CRYPT’ 91, Lect. Notes in Comput. Sci., 547, 1991, 378-386 · Zbl 0766.94012 |
| [10] | Tokareva N. N., Nelineinye bulevy funktsii: bent-funktsii i ikh obobscheniya, LAP, Saarbrucken, 2011, 180 pp. |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
