A study on maximum distance separable matrices with their applications

Abstract

In our digitally connected world we share a lot of personal information and classified data through insecure channels which require robust protection against third-party threats. Thus, establishing secure communication channels becomes imperative and block ciphers emerge as key guardians of confidentiality, integrity, and authenticity in this digital landscape. The use of Maximum Distance Separable (MDS) matrices in block cipher design plays a crucial role in defending against various attacks, and this thesis delves into the intricate world of MDS matrices. MDS matrices trace their origins to the generator matrix of maximum distance separable codes in coding theory — a code that achieves the Singleton bound. Stemming from the most fascinating code of coding theory and finding applications in symmetric key cryptography schemes, MDS matrices have garnered substantial attention due to their various direct constructions, recursive constructions, and lightweight constructions. Each methodofconstructingMDSmatrices unfolds its significance, creating a vibrant landscape for independent research. The initial part of this thesis specifically emphasizes the direct construction of MDS matrices and introduces easily implementable strategies for their inverse matrices. This research endeavor began in 1977 with the proposition by Macwillams and Solane that utilizes Cauchy matrices over finite fields for the direct construction of MDS matrices. Following this result, we introduce a new construction for MDS matrices which are not involutory, but semi-involutory in nature. These findings open up a new avenue in the construction of easily invertible MDS matrices, considering the generalization of both involutory and orthogonal properties. We have demonstrated that several Cauchy based constructions proposed by Youssef, Mister and Tavares, Gupta and Ray, while not inherently involutory or orthogonal, can have their inverse matrices easily implemented by utilizing the original matrix and multiplying it with specific diagonal matrices. In this thesis, we study another significant category of matrices– circulant matrices. Our initial focus involves examining the characteristics of the associated diagonal matrices of a circulant semi-involutory (semi-orthogonal) matrix over finite fields. Next, our attention turns to the diverse generalizations of circulant matrices. Specifically, we explore two prominent types: g-circulant matrices, introduced by Friedman in 1961, and cyclic matrices, which were introduced by Liu and Sim in 2016. We establish a profound connection between these two matrices and leveraging this connection, we provide a positive resolution to the conjecture posited by Liu and Sim. Infact, we prove the non-existence of involutory g-circulant MDS matrices of order 2d×2d over the finite field F2m. A thorough exploration into g-circulant MDS matrices is conducted, considering properties such as involutory, orthogonal, semi-involutory, and semi-orthogonal. We also present a comprehensive exploration of the general structure of semi-involutory maximum distance separable matrices of order of 3×3 over finite fields of characteristic 2. Our findings align with the research conducted on involutory MDS matrices by G¨uzel, Sakalli, Akleylek, Rijmen and C¸engellenmis¸ and some other authors. These generalized structures provide valuable insights into the overall count of MDS matrices across finite fields. Notably, for orders exceeding four, the pursuit of such structures remains an open avenue of investigation. In the last part of the thesis, we revisit a generalization of conventional encryption schemes known as Format Preserving Encryption (FPE) schemes. Traditional encryption techniques inherently mandate the elimination of the input format to maintain the “semantic security” of the encryption algorithm. However, there arise scenarios where it becomes imperative to not only retain the format but also preserve the length of the plaintext. This capability proves valuable in practical applications, such as encrypting sensitive information like credit card numbers, social security numbers, or database entries, where maintaining the original structure is crucial. Note that, a standard block cipher wouldrequire a fixedsize input andproducea(possibly longer than the plaintext) f ixed size output. This gap between what was available and what was needed in certain practical situations prompted the exploration and design of encryption schemes that preserve both the length and format of the input. The first formal study of such schemes, known as Format Preserving Encryption schemes, was initiated by Bellare et al. in 2009. Since then, numerous FPE schemes have been proposed by various authors up to the present day. In the year 2016, Gupta et al. defined an algebraic structure named Format Preserving Set (FPS) in the diffusion layer of an FPE scheme. Their work established a significant correlation between the cardinality of these sets and the potential message space of an FPE scheme over a finite field. This result affirms that numerous crucial cardinalities within the message space are unattainable over finite fields. Subsequently, Barua et al. extended the search of FPS over finite commutative rings. Building upon this generalization, we present diverse constructions of format preserving sets over f inite commutative rings with identity and finite modules over principal ideal domains. Notably, we provide examples of format preserving sets with cardinalities of 26 and 52 over torsion modules and rings. These particular cardinalities hold significance as they align with the sets of English alphabets, both in lowercase and with capitalization. Moreover, by considering a finite Abelian group as a torsion module over a PID, we show that a matrix M with entries from the PID is MDS if and only if M is MDS under the projection map on the same Abelian group.