APN Fonksiyonlar ile Tanımlanan Bazı İkili Yarı-mükemmel Lineer Kodlar

Yıl 2022, , 514 - 519, 31.08.2022

Seher Tutdere

https://doi.org/10.31590/ejosat.1057393

Öz

2021 yılında, Tutdere, minimum uzaklığı bir tek l sayısından büyük olan bir primitif ikili devirsel kodlar sınıfının örtme yarıçapı R nin r≤R≤l eşitsizliğini sağladığını göstermiştir, burada l, r verilen koda bağlı olan tam sayılardır. Burada, ilk olarak kuadratik APN (hemen hemen mükemmel lineer olmayan) fonksiyon olan Gold fonksiyonlar ile tanımlanan lineer kodların denklikleri incelenmiştir. Daha sonra Tutdere’nin elde ettiği sonucun uygulanarak bu yarı-mükemmel kodların örtme yarıçaplarının hesaplanabileceği gösterilmiştir. 2016 yılında Li ve Helleseth, kuadratik APN fonksiyonlar ile tanımlanan lineer kodların yarı-mükemmel olduklarını göstermişlerdir ve kuadratik olmayan APN fonksiyonlar ile tanımlanan kodların yarı-mükemmel olup olmadığı problemini sunmuşlardır. Burada, sonlu cisim F_(2^m ) , 1≤m≤8 için, üzerinde tanımlanan kuadratik olmayan bazı APN fonksiyonlar ile tanımlanan lineer kodların örtme yarıçapları hesaplanarak bu kodların yarı-mükemmel olduğu gösterilmiştir.

Anahtar Kelimeler

APN fonksiyonlar, Sonlu Cisim, Örtme Yarıçapı, Devirli Kodlar.

Some Binary Quasi-perfect Linear Codes Defined by APN Functions

Öz

In 2021, Tutdere proved that the covering radii R of a class of primitive binary cyclic codes with minimum distance strictly greater than an odd integer l satisfy r≤R≤l, where l, r are some integers depending on the given code. We here first discuss some equivalences of linear codes defined by Gold functions, which are quadratic APN (almost perfect nonlinear) functions. We then show that by applying the result of Tutdere one can find the covering radii of these quasi-perfect codes. In 2016, Li and Helleseth proved that the linear codes defined by the quadratic APN functions are quasi-perfect and they asked whether the linear codes defined by the non-quadratic APN functions are quasi-perfect or not. We here prove that the linear codes defined by some non-quadratic APN functions over the finite field〖 F〗_(2^m ) , for 1≤m≤8, are quasi-perfect, by computing the covering radii of these codes.

Anahtar Kelimeler

APN functions, Finite field, Covering radius, Cyclic code

Kaynakça

• Canteaut, A., P. Charpin, P., & Dobbertin, H. (2000). Binary m-sequences with three-valued crosscorrelation: a proof of Welch’s conjecture. IEEE Trans. Inf. Theory, 46(1), 4-8.
• Carlet, C. (2010). Vectorial Boolean functions for cryptography. In Boolean Models and Methods in Mathematics. Computer Science, and Engineering. Eds. Y. Crama and P. L. Hammer, Cambridge Univ. Press, 398-469.
• Carlet, C., Charpin, P., & Zinoviev, V. (1998). Codes, bent functions and permutations suitable for DES-like cryptosystems. Des. Codes. Crypt., 15(2), 125-156.
• Cohen, G. D., Honkala, I., Litsyn, S., & Lobstein, A. (1997). Covering Codes. Elsevier.
• Cohen, G. D., Karpovsky, M. G., Jr. Mattson, H. F., and Schatz, J. R. (1985). Covering radius-survey and recent results. IEEE Trans. Inform. Theory, 31(3), 328-343.
• Cohen, G. D., Litsyn, S. N., Lobstein, A. C., & Jr. Mattson, H. F. (1997). Covering radius 1985-1994. Appl. Algebra Engrg. Comm. Comput., 8(3), 173-239.
• Çalışkan, B. (2021). Z_8+𝑢Z_8+𝑣Z_8 Üzerinde Aykırı Devirli Kodlar İçin Bazı Sonuçlar. Avrupa Bilim ve Teknoloji Dergisi, (28), 660-664.
• Delsarte, P. (1973). Four fundamental parameters of a code and their combinatorial significance. Inf. Control, 23, 407- 438.
• Dobbertin, H. (1999). Almost perfect nonlinear power functions on GF(2n): the Welch case. IEEE Trans. Inf. Theory, 45(4), 1271-1275.
• Dobbertin, H. (1999). Almost perfect nonlinear power functions on GF(2n): the Niho case. Inf. Comput., 151(1), 57-72.
• Dobbertin, H. (2001). Almost perfect nonlinear power functions on GF(2n): a new case for n divisible by 5. Finite Fields and Applications, Springer, Berlin, Heidelberg. 113- 121. https://doi.org/10.1007/978-3-642-56755-1_11
• Gold, R. (1968). Maximal recursive sequences with three- valued recursive cross-correlation functions. IEEE Trans. Inf. Theory, 14(1), 154-156.
• Helleseth, T. (1985). On the covering radius of cyclic linear codes and arithmetic codes. Discrete Appl. Math., 11(2), 157-173.
• Hollmann, H. & Xiang, Q. (2001). A proof of the Welch and Niho conjectures on cross-correlations of binary m- sequences. Finite Fields and Their Applications, 7(2), 253- 286.
• Kavut, S. & S. Tutdere, S. (2019). The covering radii of a class of binary cyclic codes and some BCH codes. Des. Codes Cryptogr., 87, 317-325.
• Kasami, T. (1971).The weight enumerators for several classes of subcodes of the 2nd order binary Reed-Muller codes. Inf. Control, 18(4), 369-394.
• Nyberg, K. (1994). Differentially uniform mappings for cryptography. Advances in Cryptology- Eurocrypt’93, LNCS 765. Springer, Berlin Heidelberg, 55-64.
• Li, C. & Helleseth, T. (2016). Quasi-perfect linear codes from planar and APN functions. Cryptography and Communications, 8(2), 215-227.
• Moreno, O & Castro, N. F. (2003). Divisibility properties for covering radius of certain cyclic codes. IEEE Trans. Inform. Theory, 49(12), 3299-3303.
• Tutdere, S. (2021). On the covering radii of a class of binary primitive cyclic codes. Hacettepe Journal of Mathematics and Statistics, DOI: 10.15672/hujms.881649.
• Van Lint, J. H. & R. Wilson, R. (1986). On the minimum distance of cyclic codes. IEEE Trans. Inform. Theory, 32(1), 23-40.
• Developers, T. S., Stein, W., Joyner, D., Kohel, D., Cremona, J., & Eröcal, B. (2020). SageMath, version 9.0. Retrieved from http://www.sagemath.org