A Key Scheduling Algorithm Based on Dynamic Quasigroup String Transformation and All-Or-Nothing Key Derivation Function

Cryptographic ciphers depend on how quickly the key affects the output of the ciphers (ciphertext). Keys are traditionally generated from small size input (Seed) to a bigger size random key. Key scheduling algorithm (KSA) is the mechanism that generates and schedules all sub-keys for each round of encryption. Researches have suggested that sub-keys should be generated separately to avoid related-key attack. Similarly, the key space should be disproportionately large to resist any attack meant for secret keys. To archive that, some algorithms adopt the use of matrixes such as quasigroup, Hybrid cubes and substitution box (S-box) to generate the encryption keys. Quasigroup has other algebraic property called “Isotopism”, which literally means Different quasigroups that has the same order of elements but different arrangements. This paper proposed a Dynamic Key Scheduling Algorithm (KSA) using Isotope of a quasigroup as the dynamic substitution table. The proposed algorithm is a modification and upgrade to Allor-nothing Key Derivation Function (AKDF). To minimize the complexity of the algorithm, a method of generating Isotope from a non-associative quasigroup using one permutation is achieved. To validate the findings, non-associativity of the generated isotopes has been tested and the generated isotopes appeared to be non-associative. Furthermore, the proposed KSA algorithm will be validated using the Randomness test proposed and recommended by NIST, Avalanche and Correlation Assessment test

Dynamic key scheduling algorithm for block ciphers using quasigroup string transformation

Cryptographic ciphers depend on how quickly the key affects the output of the ciphers (ciphertext). Keys are traditionally generated from small size input (seed) to a bigger size random key(s). Key scheduling algorithm (KSA) is the mechanism that generates and schedules all sub-keys for each round of encryption. Researches have suggested that sub-keys should be generated separately to avoid related-key attack. Similarly, the key space should be disproportionately large to resist any attack on the secret key. To archive that, some algorithms adopt the use of matrixes such as quasigroup, Hybrid cubes and substitution box (S-box) to generate the encryption keys. Quasigroup has other algebraic property called “Isotophism”, which literally means Different quasigroups that has the same order of elements but different arrangements can be generated from the existing one. This research proposed a Dynamic Key Scheduling Algorithm (KSA) using isotope of a quasigroup as the dynamic substitution table. A method of generating isotope from a non-associative quasigroup using one permutation with full inheritance is achieved. The generic quasigroup string transformation has been analyzed and it is found to be vulnerable to ciphertext only attack which eventually led to the proposal of a new quasigroup string transformation in this research to assess its strength as it has never been analyzed nor properly implemented before. Based on the dynamic shapeless quasigroup and the proposed new string transformation, a Dynamic Key Scheduling Algorithm (DKSA) is developed. To validate the findings, non-associativity of the generated isotopes has been tested and the generated isotopes appeared to be non-associative. Furthermore, the proposed KSA algorithm has been validated using the randomness test proposed and recommended by NIST, avalanche test and has achieved remarkable result of 94%, brute force and correlation assessment test with -0.000449 correlations. It was fully implemented in a modified Rijndael block cipher to validate it performance and it has produced a remarkable result of 3.35332 entropy

A historical perspective of the theory of isotopisms

In the middle of the twentieth century, Albert and Bruck introduced the theory of isotopisms of non-associative algebras and quasigroups as a generalization of the classical theory of isomorphisms in order to study and classify such structures according to more general symmetries. Since then, a wide range of applications have arisen in the literature concerning the classification and enumeration of different algebraic and combinatorial structures according to their isotopism classes. In spite of that, there does not exist any contribution dealing with the origin and development of such a theory. This paper is a first approach in this regard.Junta de Andalucí

Distribución de álgebras de lie, MALCEV y evolución en clases de isotopismos

El presente manuscrito trata distintos aspectos de la teoría de isotopismos de álgebras, centrándose en particular en los isotopismos de álgebras de Lie, de Malcev y de evolución, los cuáles no han sido suficientemente estudiados en la literatura. La distribución que sigue el manuscrito se detalla a continuación. En el Capítulo 1 se expone un breve estudio acerca del origen y desarrollo de la teoría de isotopismos, constituyendo en este sentido la primera introducción en la literatura existente en introducir la mencionada teoría desde un punto de vista general. El Capítulo 2 trata de aquellos resultados en Geometría Algebraica Computacional y en Teoría de Grafos que usamos a lo largo del manuscrito con vistas a determinar computacionalmente las clases de isotopismos de cada tipo de álgebra bajo consideración en los siguientes capítulos. Se describen en particular un par de grafos que permiten definir funtores inyectivos entre álgebras de dimensión finita sobre cuerpos finitos y los citados grafos. El cálculo computacional de invariantes por isomorfismos de estos grafos juega un papel destacable en la distribución de las distintas familias de álgebras en clases de isotopismos y de isomorfismos. Algunos resultados preliminares son expuestos en este sentido, particularmente acerca de la distribución de anillos de cuasigrupos parciales sobre cuerpos finitos. El Capítulo 3 se centra en la distribución de clases de isomorfismos y de isotopismos de dos familias de álgebras de Lie: el conjunto Pn;q de álgebras de Lie prefiliformes n-dimensionales sobre el cuerpo finito Fq y el conjunto Fn(K) de álgebras de Lie filiformes n-dimensionales sobre un cuerpo K. Se prueba concretamente la existencia de n clases de isotopismos en Pn;q. También se introducen dos nuevas series de invariantes por isotopismos que son usados para determinar las clases de isotopismos del conjunto Fn(K) para n≤7 sobre cuerpos algebraicamente cerrados y sobre cuerpos finitos. El Capítulo 4 trata con distintos ideales radicales cero-dimensionales cuyos conjuntos algebraicos asociados pueden indentificarse de forma única con el conjunto Mn(K) de álgebras de Malcev n-dimensionales sobre un cuerpo finito K. El cálculo computacional de sus bases reducidas de Gröbner, junto a la clasificación de álgebras de Lie sobre cuerpos finitos dada por De Graaf y Strade, permiten determinar la distribución de M3(K) y M4(K) no sólo en clases de isomorfismos, que es el criterio usual, sino también en clases de isotopismos. En concreto, probamos la existencia de cuatro clases de isotopismos en M3(K) y ocho clases de isotopismos en M4(K). Además, se prueba que todo álgebra de Malcev 3-dimensional sobre cualquier cuerpo finito y todo álgebra de Malcev 4-dimensional sobre un cuerpo finito de característica distinta de dos es isotópica a un magma-álgebra de Lie. Finalmente, el Capítulo 5 trata con el conjunto En(K) de álgebras de evolución n-dimensionales sobre un cuerpo K, cuya distribución en clases de isotopismos está relacionada de forma única con mutaciones en Genética no Mendeliana. Se centra en concreto en el caso bi-dimensional, el cuál está relacionado con los procesos de reproducción asexual de organismos diploides. Se prueba en particular que el conjunto E2(K) se distribuye en cuatro clases de isotopismos, independientemente de cuál sea el cuerpo base y se caracteriza sus clases de isomorfismos.This manuscript deals with distinct aspects of the theory of isotopisms of algebras. Particularly, we focus on isotopisms of Lie, Malcev and evolution algebras, for which this theory has not been enough studied in the literature. The manuscript is organized as follows. In Chapter 1 we expose a brief survey about the origin and development of the theory of isotopisms. This constitutes a first attempt in the literature to introduce this theory from a general point of view. Chapter 2 deals with those results in Computational Algebraic Geometry and Graph Theory that we use throughout the manuscript in order to compute the isotopism classes of each type of algebra under consideration in the subsequent chapters. We describe in particular a pair of graphs that enable us to define faithful functors between finite-dimensional algebras over finite fields and these graphs. The computation of isomorphism invariants of these graphs plays a remarkable role in the distribution of distinct families of algebras into isotopism and isomorphism classes. Some preliminary results are exposed in this regard, particularly on the distribution of partial-quasigroup rings over finite fields. Chapter 3 focuses on the distribution into isomorphism and isotopism classes of two families of Lie algebras: the set Pn;q of n-dimensional pre- filiform Lie algebras over the finite field Fq and the set Fn(K) of n-dimensional filiform Lie algebras over a base field K. Particularly, we prove the existence of n isotopism classes in Pn;q. We also introduce two new series of isotopism invariants that are used to determine the isotopism classes of the set Fn(K) for n ≤ 7 over algebraically closed fields and finite fields. Chapter 4 deals with distinct zero-dimensional radical ideals whose related algebraic sets are uniquely identified with the set Mn(K) of n-dimensional Malcev magma algebras over a finite field K. The computation of their reduced Gröbner bases, together with the classification of Lie algebras over finite fields given by De Graaf and Strade, enable us to determine the distribution of M3(K) and M4(K) not only into isomorphism classes, which is the usual criterion, but also into isotopism classes. Particularly, we prove the existence of four isotopism classes in M3(K) and eight isotopism classes in M4(K). Besides, we prove that every 3-dimensional Malcev algebra over any finite field and every 4-dimensional Malcev algebra over a finite field of characteristic distinct from two is isotopic to a Lie magma algebra. Finally, Chapter 5 deals with the set En(K) of n-dimensional evolution algebras over a field K, whose distribution into isotopism classes is uniquely related with mutations in non-Mendelian genetics. Particularly, we focus on the two-dimensional case, which is related to the asexual reproduction processes of diploid organisms. We prove that the set E2(K) is distributed into four isotopism classes, whatever the base field is, and we characterize its isomorphism classes