Units in quasigroups with Bol-Moufang type identities Natalia N. - - PowerPoint PPT Presentation

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Units in quasigroups with Bol-Moufang type identities

Natalia N. Didurik∗ and Victor A. Shcherbacov⋆

Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Budapest, July 11, 2019

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects

Groupoids and quasigroups

Groupoids

Definition

A binary groupoid (G, A) is a non-empty set G together with binary operation A. We give main definition of a quasigroup:

Definition

Binary groupoid (Q, ◦) is called a quasigroup if for any ordered pair (a, b) ∈ Q2 there exist unique solutions x, y ∈ Q to the equations x ◦ a = b and a ◦ y = b [2]. Often this definition is called existential. In fact this definition was given by Ruth Moufang.

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects

Units

Definitions

Definition

◮ An element f ∈ Q is a left unit of (Q; ·) iff f · x = x for all

x ∈ Q.

◮ An element e ∈ Q is a right unit of (Q; ·) iff x · e = x for all

x ∈ Q.

◮ An element s ∈ Q is a middle unit of (Q; ·) iff x · x = s for all

x ∈ Q.

◮ A quasigroup which has left unit is called here left loop. ◮ A quasigroup which has right unit is called here right loop. ◮ A quasigroup (Q, ·) with an identity element e ∈ Q is called a

loop.

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects

Introduction

An algebra with three binary operations (Q, ·, /, \) that satisfies the following identities: x · (x\y) = y, (1) (y/x) · x = y, (2) x\(x · y) = y, (3) (y · x)/x = y, (4) is called a quasigroup (Q, ·, /, \) [2]. Basic facts about quasigroups and loops can be found in [2].

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects

Definition

Let (G, ·) be a groupoid and let a be a fixed element in G. Translation maps La (left) and Ra (right) are defined by Lax = a · x, Rax = x · a for all x ∈ G. For quasigroups it is possible to define a third kind of translation, namely, middle translations. If Pa is a middle translation of a quasigroup (Q, ·), then x · Pax = a for all x ∈ Q [3].

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects

Belousov problems

Belousov’s Problem #18. How to recognize identities which force quasigroups satisfying them to be loops [2]? In the first place we note the result of J.D.H. Smith [26, Proposition 1.3]. A nonempty quasigroup (Q, ·, /, \) is a loop if and only if it satisfies the “slightly associative identity” x(y/y) · z = x · (y/y)z. (5) You understand that previous theorem does not solve Belousov problem, but this is an important criterion. Notice, identity (5) is the generalized Bol-Moufang type identity in the language of this presentation,

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects

Generalized Belousov problems

Generalized Belousov’s Problem #18. How to recognize identities which force quasigroups satisfying them to have left, right, middle unit [17]? In the article [17] we try to research generalized Belousov’s Problem #18 for quasigroups with identities of two variables and

• ne fixed element. For example, we research which units has

quasigroup (Q, ·) with the following identity and with fixed element a: ax · y = a · xy. (6) Here letters x, y denote free variables, and the letter a denotes a fixed element of the set Q.

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects

Belousov problems

G.B. Belyavskaya [5, 6] studied identities with permutations. An identity with permutations can be defined as an equality between two terms that contain variables and symbols of quasigroup operations ·, /, \ and permutations of the set Q incorporated in this equality. We can re-write identity (6) in the following form: Lax · y = La(xy). (7) So identity (7) is an identity with permutation. Implicitly this concept is well known in quasigroup theory. For example, identities with permutations are used in the definition of an IP-loop [R. Moufang, Math. Ann. 110 (1935), no. 1, 416–430;].

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects

Belousov problems

It is clear that we can see on the equality (6) as on an autotopy (La, ε, La) of a quasigroup (Q, ·). We can say that equality (6) defines the left nuclear element a. Moreover, we can rewrite equality (6) in the form ax · y = a · (x ◦ y). (8) In this case we obtain right derivative groupoid (operation) (Q, ◦)

• f quasigroup (Q, ·) [2, 24, 17].

In the article of Alexandar Krapez [16] generalized Belousov problem is researched for groupoids using generalized derivatives and functional equations.

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal Objects

Theorem

Autotopy f e s Autotopy f e s (La, La, ε) + − − (La, ε, La) + − − (La, L−1

a , ε)

+ − − (La, ε, L−1

a )

+ − − (La, Ra, ε) − − − (La, ε, Ra) + − − (La, R−1

a , ε)

− − − (La, ε, R−1

a )

+ − − (La, Pa, ε) + − − (La, ε, Pa) − − − (La, P−1

a , ε)

+ − − (La, ε, P−1

a )

− − −

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Definitions

An identity based on a single binary operation is of Bol-Moufang type if “both sides consist of the same three different letters taken in the same order but one of them occurs twice on each side”[9]. We use list of 60 Bol-Moufang type identities given in [11]. There exist other (“more general”) definitions of Bol-Moufang type identities and, therefore, other lists and classifications of such identities [1, 8].

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Definitions

We recall, (12)-parastroph of groupoid (G, ·) is a groupoid (G, ∗) such that the operation “∗ ” is obtained by the following rule: x ∗ y = y · x. (9) It is clear that for any groupoid (G, ·) there exists its (12)-parastroph groupoid (G, ∗). Suppose that an identity F is true in groupoid (G, ·). Then we can

• btain (12)-parastrophic identity (F ∗) of the identity F replacing

the operation “·” with the operation “∗” and changing the order

• f variables following rule (9).

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

(12)-parastrophes of identities

Theorem

(F1)∗ = F3, (F2)∗ = F4, (F5)∗ = F10, (F6)∗ = F6, (F7)∗ = F8, (F9)∗ = F9, (F11)∗ = F24, (F12)∗ = F23, (F13)∗ = F22, (F14)∗ = F21, (F15)∗ = F30, (F16)∗ = F29, (F17)∗ = F27, (F18)∗ = F28, (F19)∗ = F26, (F20)∗ = F25, (F31)∗ = F34, (F32)∗ = F33, (F35)∗ = F40, (F36)∗ = F39, (F37)∗ = F37, (F38)∗ = F38, (F41)∗ = F53, (F42)∗ = F54, (F43)∗ = F51,(F44)∗ = F52, (F45)∗ = F60, (F46)∗ = F56, (F47)∗ = F58, (F48)∗ = F57, (F49)∗ = F59, (F50)∗ = F55. In quasigroup case similarly to (12)-parastrophe identity other parastrophe identities can be defined and studied.

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Middle unit

Easy to see that any group satisfies any of identities F1 − F60. Therefore the cyclic group Z3 is a counter-example to proposition that “there exist a quasigroup with an identity from list of identities F1 − F60 and which has middle unit”.

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Left and right units, groups

N. Ab. Id. f e Lo. Gr. F1 xy · zx = (xy · z)x + + + + F2

• m. Mou.

xy · zx = (x · yz)x + + + − F3 xy · zx = x(y · zx) + + + + F4

• m. Mou.

xy · zx = x(yz · x) + + + − F5 (xy · z)x = (x · yz)x + + + + F6 extra id. (xy · z)x = x(y · zx) + + + − F7 (xy · z)x = x(yz · x) + − − − F8 (x · yz)x = x(y · zx) − + − −

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Left and right units, groups

N. Ab. Id. f e Lo. Gr. F9 (x · yz)x = x(yz · x) − − − − F10 x(y · zx) = x(yz · x) + + + + F11 xy · xz = (xy · x)z + + + + F12 xy · xz = (x · yx)z + + + + F13 extra identity xy · xz = x(yx · z) + + + − F14 xy · xz = x(y · xz) + + + + F15 (xy · x)z = (x · yx)z − − − − F16 (xy · x)z = x(yx · z) + − − −

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Left and right units, groups

N. Ab. Id. f e Lo. Gr. F52 yz · xx = (y · zx)x − + − − F53 RC identity yz · xx = y(zx · x) + + + − F54 yz · xx = y(z · xx) − + − − F55 (yz · x)x = (y · zx)x + + + + F56 RC identity (yz · x)x = y(zx · x) − − − − F57 RC identity (yz · x)x = y(z · xx) − + − − F58 (y · zx)x = y(zx · x) + + + + F59 (y · zx)x = y(z · xx) − + − − F60 y(zx · x) = y(z · xx) − + − −

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Introduction

Moufang identities

Definition

Let (Q, ·) be a groupoid. The following identities ((x · yz)x = xy · zx, x(yz · x) = xy · zx, x(y · xz) = (xy · x)z, (zx · y)x = z(x · yx)) are called Moufang identities. A quasigroup (Q, ·) with any Moufang identity is a loop [10, 18, 23]. The easiest way to define commutative Moufang loop (Q, ·) is to take one of Moufang identities and add identity of commutativity.

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Introduction

Definitions

R.H. Bruck [7] has given standard definition of a CML (Q, ·) as a loop (Q, ·) with the following left semimedial identity xx · yz = xy · xz ⇔ x2 · yz = xy · xz. (10) See also [2, 20]. Identity (10) is a partial case (x = y) of medial identity xy · uv = xu · yv. Left semimedial idempotent quasigroup satisfies left distributive identity x · yz = xy · xz.

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Introduction

Definitions

The following identity xy · zz = xz · yz ⇔ xy · z2 = xz · yz (11) is called right semimedial. This identity can also be used to define a CML. Classes of quasigroups with identities (10) and (11) are studied in [12, 14, 13, 25, 15, 21].

Example

We give an example of left semimedial quasigroup. ∗ 1 2 1 2 1 1 2 2 2 1

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Results

Theorem

Any of the following identities (Cote, Manin) x(xy · z) = (z · xx)y, (12) x(y · xz) = (x2 · y)z (13) defines the class of commutative Moufang loops in the class of loops [8, 19].

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Results

Example

The following quasigroup satisfies identities (12), (13) and does not have any left and right identity element. ∗ 1 2 2 1 1 2 1 2 1 2

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Results

Theorem

Any of the following identities (xy · x)z = (y · xz)x, (14) x(y · zy) = y(xy · z) (15) defines the class of commutative Moufang loops in the class of quasigroups.

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Quasigroup based El Gamal encryption system

We give an analogue of El Gamal encryption system based on Markovski algorithm [22]. Let (Q, f ) be a binary quasigroup and T = (α, β, γ) be its isotopy. Alices keys are as follows: Public Key is (Q, f ), T, T (m,n,k) = (αm, βn, γk), m, n, k ∈ N, and Markovski algorithm. Private Key m, n, k.

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Quasigroup based El Gamal encryption system

Encryption To send a message b ∈ (Q, f ) Bob computes T (r,s,t), T (mr,ns,kt) for a random r, s, t ∈ N and (T (mr,ns,kt)(Q, f )). The ciphertext is (T (r,s,t), T (mr,ns,kt)(Q, f ), (T (mr,ns,kt)(Q, f ))b). To obtain (T (mr,ns,kt)(Q, f ))b Bob uses Markovski algorithm which is known to Alice.

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Quasigroup based El Gamal encryption system

Decryption Alice knows m, n, k, so if she receives the ciphertext (T (r,s,t), T (mr,ns,kt)(Q, f ), (T (mr,ns,kt)(Q, f ))b), she computes T (−rm,−ns,−kt) from T (r,s,t) and then (Q, f ), further she computes (Q, f )−1 and, finally, she computes b.

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

Quasigroup based El Gamal encryption system

In this algorithm an isostrophy [21] can also be used instead of isotopy, more complicate Algorithm instead of Markovski algorithm, n-ary (n > 2) quasigroups [4, 22] instead of binary quasigroups.

Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

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Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

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Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

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Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

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Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

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Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

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Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆

Introduction Units in quasigroups with classical Bol-Moufang identities Identities defining a CML Results Quasigroup based El Gamal

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Units in quasigroups with Bol-Moufang type identities Dimitrie Cantemir State University∗ and Institute of Mathematics and Computer Science⋆