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Difference of Function on Vector Space over F

100%
Arai K., Wakabayashi K., Okazaki H.
Formalized Mathematics
|
2014
|
tom 22
|
nr 3
269-275
EN
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.
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Formalization of the Advanced Encryption Standard. Part I

80%
Arai K., Okazaki H.
Formalized Mathematics
|
2013
|
tom 21
|
nr 3
171-184
EN
In this article, we formalize the Advanced Encryption Standard (AES). AES, which is the most widely used symmetric cryptosystem in the world, is a block cipher that was selected by the National Institute of Standards and Technology (NIST) as an official Federal Information Processing Standard for the United States in 2001 [12]. AES is the successor to DES [13], which was formerly the most widely used symmetric cryptosystem in the world. We formalize the AES algorithm according to [12]. We then verify the correctness of the formalized algorithm that the ciphertext encoded by the AES algorithm can be decoded uniquely by the same key. Please note the following points about this formalization: the AES round process is composed of the SubBytes, ShiftRows, MixColumns, and AddRoundKey transformations (see [12]). In this formalization, the SubBytes and MixColumns transformations are given as permutations, because it is necessary to treat the finite field GF(28) for those transformations. The formalization of AES that considers the finite field GF(28) is formalized by the future article.
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