APPIAH, F. Public-key cryptography and Chebyshev polynomials [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2023.

Posudek vedoucího

Civino, Roberto

Francis Appiah's thesis is based on a discrete version of Chebyshev polynomials and their use in public-key cryptosystems where, starting from a commutation property, can replace standard monomials used in more classical constructions. The aim of the work is to introduce the necessary prerequisites on discrete Chebyshev polynomials and to describe two public-key cryptosystems, respectively an RSA-like and a ElGamal-like, whose security is based on the alleged intractability of computational problems related to Chebyshev polynomials. The discussion is based on some classical references and on some recent scientific papers where cryptographic schemes are introduced and examined. The thesis was not meant to contain innovative contributions and in fact it represents only an essay on the subject. However, the subject is presented with sufficient clarity and the document is sufficiently self-contained and readable. For the reasons described above, my rating of the candidate's work is B.

Posudek oponenta

Kureš, Miroslav

The presented thesis deals with a public key cryptography system based on Chebyshev polynomials. The original design of the system, which was in the quoted article from some proceedings published in 2003, was based on the properties of Chebyshev polynomials over real numbers, is itself highly questionable, since real numbers are of course absolutely unsuitable for cryptography implemented on computers. However, the author of thesis solves this in the presented work and presents new concept over finite fields.In the introductory chapters, an unnecessarily large space is devoted to elementary concepts such as group, Abelian group, RSA system, etc. In the following part, as the author starts writing about the torus, the definitions are naively vague, like "torus is a doughnut-shaped object with a hole in the middle” (p. 28); or the following "definition": Def 3.1: "An automorphism of a torus is a bijective function that preserves the algebraic and topological structure of the torus." Mathematically, it remains unclear. This is followed by a chapter about Chebyshev polynomials of the first kind, summarizing their known properties. (Again, some just lengthen the text and will not be needed in further derivations.) The inappropriateness of a cryptographic use of Chebyshev polynomials over real numbers is noted and that is why they are transferred over finite fields, where their periodic properties are observed. Property 4. 16 (p. 42) may be found crucial allowing potential use in cryptography. Two encryption schemes are then proposed, the author explains them clearly and gives examples.I note that the author has fulfilled the assignment. But his actual original work is very small. In most of the text, he repeats very well-known concepts, often very vaguely, often at a high-school level. As a final university thesis, this text is not of a significant quality. I admit, however, that the author probably understood the topic and managed to explain it.The thesis is written in comprehensible average English, there are no significant errors. I recommend the thesis for defense.

eVSKP id 150327