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Solving AX-equations
Butora, Jan ; Tůma, Jiří (advisor) ; Joščák, Daniel (referee)
Title: Solving AX-equations Author: Jan Butora Department: Department of algebra Supervisor: doc. RNDr. Jiří Tůma, DrSc., Department of algebra Abstract: In this work, we present concept of AX-equations and focus on two such equations. Using similiar techniques, we build a theory for both equations, which allows us to express number of their solutions based only on their parameters. Using this theory, we demonstrate on an example that differential steps, used in differential cryptanalysis of modular addition, are not independent. Moreover, based on this theory we introduce and implement fast algorithms for searching solutions. Keywords: differential cryptanalysis, AX-equations, modular addition, carry, sol- vability condition
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Discrete Channel Capacity
Butora, Jan ; Holub, Štěpán (advisor) ; Žemlička, Jan (referee)
Title: Discrete Channel Capacity Author: Jan Butora Department: Department of Algebra Supervisor: doc. Mgr. Štěpán Holub, Ph.D. et Ph.D., Department of Algebra Abstract: This Bachelor thesis introduces and examines C.E. Shannon's discrete channel capacity theory, which was first published in 1948 as one of the founding studies in the field of mathematical information theory. In the first place, possible way of information measurement is presented and communication systems are described. Additionally, emphasis is given to discrete noiseless channel and the theorem on calculating the capacity of such channels is examined and proven. Shannon's proof is examined in detail as it contains several non-trivial results in finite differences. Finally, calculation of channel capacity using the theorem is shown in practice. Keywords: difference equations, generating function, channel capacity
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