Difference of Function on Vector Space over F

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Summary

In [11], the definitions of forward difference, backward difference, and central difference as difference operations for functions on R were formalized. However, the definitions of forward difference, backward difference, and central difference for functions on vector spaces over F have not been formalized. In cryptology, these definitions are very important in evaluating the security of cryptographic systems [3], [10]. Differential cryptanalysis [4] that undertakes a general purpose attack against block ciphers [13] can be formalized using these definitions. In this article, we formalize the definitions of forward difference, backward difference, and central difference for functions on vector spaces over F. Moreover, we formalize some facts about these definitions.

[1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.

[2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.

[3] E. Biham and A. Shamir. Differential cryptanalysis of DES-like cryptosystems. Lecture Notes in Computer Science, 537:2-21, 1991.

[4] E. Biham and A. Shamir. Differential cryptanalysis of the full 16-round DES. Lecture Notes in Computer Science, 740:487-496, 1993.

[5] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.

[6] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.

[7] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.

[8] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.

[9] Eugeniusz Kusak, Wojciech Leonczuk, and Michał Muzalewski. Abelian groups, fields and vector spaces. Formalized Mathematics, 1(2):335-342, 1990.

[10] X. Lai. Higher order derivatives and differential cryptoanalysis. Communications and Cryptography, pages 227-233, 1994.

[11] Bo Li, Yan Zhang, and Xiquan Liang. Difference and difference quotient. Formalized Mathematics, 14(3):115-119, 2006. doi:10.2478/v10037-006-0014-z.

[12] Michał Muzalewski and Wojciech Skaba. From loops to Abelian multiplicative groups with zero. Formalized Mathematics, 1(5):833-840, 1990.

[13] Hiroyuki Okazaki and Yasunari Shidama. Formalization of the data encryption standard. Formalized Mathematics, 20(2):125-146, 2012. doi:10.2478/v10037-012-0016-y.

[14] Beata Perkowska. Functional sequence from a domain to a domain. Formalized Mathematics, 3(1):17-21, 1992.

[15] Christoph Schwarzweller. The binomial theorem for algebraic structures. Formalized Mathematics, 9(3):559-564, 2001.

[16] Wojciech A. Trybulec. Groups. Formalized Mathematics, 1(5):821-827, 1990.

[17] Wojciech A. Trybulec. Vectors in real linear space. Formalized Mathematics, 1(2):291-296, 1990.

[18] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.

[19] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.

[20] Edmund Woronowicz. Relations defined on sets. Formalized Mathematics, 1(1):181-186, 1990.

[21] Hiroshi Yamazaki and Yasunari Shidama. Algebra of vector functions. Formalized Mathematics, 3(2):171-175, 1992.

Formalized Mathematics

(a computer assisted approach)

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researchers in the fields of formal methods and computer-checked mathematics

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