TOTALLY SYMMETRIC AND MEDIAL QUASIGROUPS AND THEIR APPLICATIONS by BENJAMIN YOUNG Submitted in partial fulfillment of the requirements for the degree of Master of Science Department of Computing and Information Science CASE WESTERN RESERVE UNIVERSITY May, 2021 CASE WESTERN RESERVE UNIVERSITY SCHOOL OF GRADUATE STUDIES We hereby approve the Thesis of Benjamin Young candidate for the degree of Master of Science*. Committee Chair Dr. Harold Connamacher Committee Member Dr. David Singer Committee Member Dr. Shuai Xu Date of Defense March 25, 2021 *We also certify that written approval has been obtained for any proprietary material contained therein. Contents List of Tables iii List of Figures iv Abstract vi 1 Introduction 1 1.1 Overview . .2 1.2 Basic Definitions . .4 1.3 Totally Symmetric and Medial Quasigroups . .8 1.4 Prior Work Counting Quasigroups . 14 2 n-ary TSM Quasigroups and Trees 17 2.1 n-ary Product Trees . 18 2.2 Generalizing Etherington's Symmetry Results . 22 3 Counting Labeled Binary and n-ary TSM Quasigroups and Abelian Groups 32 3.1 Abelian Groups and Totally Symmetric Medial Quasigroups . 33 3.2 Binary and n-ary Abelian Groups . 40 3.3 Binary and n-ary TSM Quasigroups . 46 3.4 n-ary Abelian Groups and TSM Quasigroups . 51 i 3.5 The Number of Labeled Binary and n-ary TSM Quasigroups and Abelian Groups . 56 4 Applications of Quasigroups 58 4.1 Quasigroups in Cryptology . 59 4.1.1 Survey of Quasigroups in Cryptology . 59 4.1.2 Generating TSM Quasigroups . 66 4.2 TSM Quasigroups and Cubic Curves . 68 4.2.1 The Chord and Tangent Construction . 68 4.2.2 Elliptic Curves and Diffie-Hellman . 74 4.2.3 Iterated Tangents . 76 ii List of Tables 2.1 Demonstrating that τλ1,λ2 = τλ1,µτµ,λ2 τλ1,µ................ 28 3.1 Number of labeled TSM quasigroups versus number of all labeled quasi- groups over k elements . 57 4.1 Example of Vigen`erecipher . 60 iii List of Figures 1.1 Cayley table of integers mod 3 . .4 1.2 Demonstrating ab = ba in a totally symmetric quasigroup . .9 1.3 Demonstrating a(ab) = b in a totally symmetric quasigroup . 10 1.4 Demonstrating (ab)(bc) = (ac)(bd) in a medial quasigroup . 10 1.5 Example of equation tree . 12 1.6 Diagram of bijections between binary and n-ary TSM quasigroups and abelian groups . 14 2.1 Example of n-ary product tree . 19 2.2 Example of a full tree . 19 2.3 Product tree illustrating the induction step in Theorem 2 . 24 2.4 Product tree after swapping sij and sk1 ................. 25 0 2.5 Product tree after swapping u and v and sij with sk1 ......... 26 n n−1 n−1 2.6 Product tree for f f f(x1 ); y1 ; z1 ................ 28 n 2.7 Equation tree for f(s1 ) = x ....................... 29 n n 2.8 Equation tree for f(s1 ) = f(t1 )..................... 30 i0−1 n n 2.9 Equation tree for f s1 ; f(t1 ); si0+1 = si0 ............... 30 3.1 Diagram of bijections between binary and n-ary TSM quasigroups and abelian groups . 33 3.2 Bijection between TSM quasigroups and abelian groups . 34 iv 3.3 Bijection between binary and n-ary abelian groups . 40 3.4 Bijection between binary and n-ary TSM quasigroups . 46 n−2 n−1 n−1 3.5 Product tree for f 0 ; gn; f 0 ; f(0; g1 ) ............ 47 n−3 n−2 n−1 3.6 Product tree for h 0 ; gn; h 0 ; h(g1 ) ............. 48 3.7 Bijection between n-ary TSM quasigroups and n-ary abelian groups . 51 4.1 The tabula recta . 60 4.2 Example of chord and tangent multiplication on a cubic curve . 70 4.3 Example of mediality on a cubic curve . 72 v Totally Symmetric and Medial Quasigroups and their Applications Abstract by BENJAMIN YOUNG We prove some new results regarding binary and n-ary totally symmetric and medial (TSM) quasigroups and explore their applications to cubic curves and cryptography. We first generalize to the n-ary case Etherington's result that a product in binary TSM quasigroup is symmetric in factors whose depths differ by a multiple of 2 in the corresponding product tree. We then demonstrate that there are an equal number of the four following labeled structures over any finite set: abelian groups, n-ary abelian groups, TSM quasigroups, and n-ary TSM quasigroups. Next we explore applications of quasigroups in cryptography and discuss how our maps between abelian groups, TSM quasigroups, and n-ary TSM quasigroups can be used to generate and easily calculate products in quasigroups for use in cryptosystems. Finally, we discuss the TSM quasigroup of points on a cubic curve and prove some properties of iterated squaring in prime-order TSM quasigroups. vi Chapter 1 Introduction 1 1.1 Overview In this thesis we prove several new results about binary and n-ary totally symmetric and medial (TSM) quasigroups and discuss these results' implications in two primary applications of quasigroups: cryptology and cubic curves. We begin Chapter 1 by presenting definitions of TSM quasigroups, abelian groups, and the n-ary variants of both. We also introduce the concept of product trees for binary groupoids. In Section 1.4, as a preface to our work in Chapter 3, we review prior work counting quasigroups and n-ary quasigroups. In Chapter 2, generalizing a result from Ether- ington, we prove that a product in an n-ary TSM quasigroup is symmetric in factors whose depths in the product's tree representation differ by a multiple of 2 - that is, we can permute those factors without changing the value of the product. In Chapter 3 we demonstrate that there are an equal number of the four fol- lowing labeled structures over any finite set: abelian groups, n-ary abelian groups, TSM quasigroups, and n-ary TSM quasigroups. Each of Chapter 3's sections focuses on finding a bijection between two sets of algebraic constructions over a finite set G. Three of these bijections were unknown prior to this thesis. In Section 3.1 we demonstrate several bijections between the sets of binary abelian groups and binary TSM quasigroups. In Section 3.2 we give a bijection from the set of binary abelian groups to the set of n-ary abelian groups for any n. In Section 3.3 we use the theory developed in Chapter 2 to reproduce Hacker's proof of a bijection between (n−1)-ary and n-ary TSM quasigroups for every n ≥ 3, giving a bijection between binary TSM quasigroups and n-ary TSM quasigroups for any n. Finally in Section 3.4 we use the results of the previous sections to derive a bijection between n-ary abelian groups and n-ary TSM quasigroups. In Chapter 4 we discuss the significance of our results to two areas of applications of TSM quasigroups. In Section 4.1 we review previous applications of quasigroups to cryptology. We discuss how the maps between abelian groups, TSM quasigroups, and 2 n-ary TSM quasigroups discussed in Chapter 3 let us easily generate binary and n- ary TSM quasigroups for use in cryptosystems, and let us calculate products in those quasigroups without having to store their Cayley tables. In Section 4.2 we define cubic plane curves and relay Etherington's proof that the points on a cubic plane curve form a TSM quasigroup under the chord and tangent construction. Finally, we discuss elliptic curve cryptography and the Diffie-Hellman key exchange, prove some results about iterated squaring in a prime-order TSM quasigroup, or equivalently iterated tangents in a cubic curve, and discuss the merits of iterated squaring as a possible operation for a Diffie-Hellman variant. 3 1.2 Basic Definitions Let G be a set with n elements, which we will usually denote by 0; 1; : : : ; n − 1. If f is any operation that takes elements of G as arguments and returns an element of G, write (G; f) to mean G equipped with the operation f. If f : Gn = G × G × ::: × G ! G | {z } n (f takes a list of n elements on G and returns an element of G), we say f is an n-ary operation. We usually denote 2-ary (binary) operations by symbols - typically +; ·; ∗, or ◦ - and frequently leave out the · and write a · b = ab. We denote n-ary operations for n > 3 as letters and use function notation, for example f(a; b; c). If (G; ·) is a set equipped with a binary operation (a magma), then the Cayley Table of (G; ·) is analogous to its multiplication table. For example, if G = f0; 1; 2g and a +3 b := a + b mod 3 then (G; +3) has the following Cayley table: 0 1 2 0 0 1 2 1 1 2 0 2 2 0 1 Figure 1.1: Cayley table of (G; +3) (G; ·) is a quasigroup if for any a; b 2 G, the equations a · x = b and y · a = b have unique solutions for x and y, respectively. The former means that if we are searching for a b in row a of (G; ·)'s Cayley table, we will find it exactly once: in column x. Similarly, the latter means that if we are searching for a b in column a we will find it exactly once: in row y. Thus every row and every column of (G; ·)'s Cayley table is a permutation of the elements of G. In other words, the Cayley table of any quasigroup (G; ·) is a Latin square.