# On some Constructions of Shapeless Quasigroups

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1 Aleksandra Mileva 1 and Smile Markovski 2 1 Faculty of Computer Science, University Goce Delčev, Štip 2 Faculty of Computer Science and Computer Engineering, University Ss. Cyril and Methodius - Skopje Republic of Macedonia Loops 11 July 25-27, Třešt, Czech Republic

2 Motivation Today, we can already speak about quasigroup based cryptography, because the number of new defined cryptographic primitives that use quasigroups is growing up: stream cipher EDON-80 (Gligoroski et al. (estream 2008)), hash functions EDON-R (Gligoroski et al. (SHA )) and NaSHA (Markovski and Mileva (SHA )), digital signature algorithm MQQ-DSA (Gligoroski et al. (ACAM 2008)), public key cryptosystem LQLP-s (for s {104, 128, 160}) (Markovski et al. (SCC 2010)), etc.

3 Motivation Different authors seek quasigroups with different properties. Definition (Gligoroski et al (2006)) A quasigroup (Q, ) of order r is said to be shapeless iff it is non-idempotent, non-commutative, non-associative, it does not have neither left nor right unit, it does not contain proper sub-quasigroups, and there is no k < 2r such that identities of the kinds x (x (x y)) = y, y = ((y x) x) x } {{ } } {{ } k k (1)

4 1 2 3 Different generalizations of the Feistel Network as orthomorphisms 4 5

5 1 2 3 Different generalizations of the Feistel Network as orthomorphisms 4 5

6 Complete mappings and orthomorphisms Definition (Mann (1949), Johnson et al (1961), Evans (1989)) A complete mapping of a quasigroup (group) (G, +) is a permutation φ : G G such that the mapping θ : G G defined by θ(x) = x + φ(x) (θ = I + φ, where I is the identity mapping) is again a permutation of G. The mapping θ is the orthomorphism associated to the complete mapping φ. A quasigroup (group) G is admissible if there is a complete mapping φ : G G.

7 Complete mappings and orthomorphisms If θ is the orthomorphism associated to the complete mapping φ, then φ is the orthomorphism associated to the complete mapping θ, The inverse of the complete mapping (orthomorphism) is also a complete mapping (orthomorphism) (Johnson et al (1961)), Two orthomorphisms θ 1 and θ 2 of G are said to be orthogonal if and only if θ 1 θ 1 2 is an orthomorphism of G too. Even more, every orthomorphism is orthogonal to I and θ 1 is orthogonal to θ if and only if θ 2 is an orthomorphism (Johnson et al (1961))

8 Sade s Diagonal method (Sade (1957)) Consider the group (Z n, +) and let θ be a permutation of the set Z n, such that φ(x) = x θ(x) is also a permutation. Define an operation on Z n by: x y = θ(x y) + y (2) where x, y Z n. Then (Z n, ) is a quasigroup (and we say that (Z n, ) is derived by θ).

9 Generalization Theorem Let φ be a complete mapping of the admissible group (G, +) and let θ be an orthomorphism associated to φ. Define operations and on G by x y = φ(y x) + y = θ(y x) + x, (3) x y = θ(x y) + y = φ(x y) + x, (4) where x, y G. Then (G, ) and (G, ) are quasigroups, opposite to each other, i.e., x y = y x for every x, y G.

10 Quasigroup conjugates Theorem Let φ : G G be a complete mapping of the abelian group (G, +) with associated orthomorphism θ : G G. Then all the conjugates of the quasigroup (G, ) can be obtain by the equation (4) and the following statements are true. a) The quasigroup (G, /) is derived by the orthomorphism θ 1 associated to the complete mapping φθ 1. b) The quasigroup (G, \) is derived by the orthomorphism θ( φ) 1 associated to the complete mapping ( φ) 1. c) The quasigroup (G, //) is derived by the orthomorphism φ( θ) 1 associated to the complete mapping ( θ) 1. d) The quasigroup (G, \\) is derived by the orthomorphism φ 1 associated to the complete mapping θφ 1. e) (G, ) = (G, ).

11 1 2 3 Different generalizations of the Feistel Network as orthomorphisms 4 5

12 If θ(0) 0 then the quasigroup (G, ) has no idempotent elements. (G, ) does not have a left unit and if θ is different to the identity mapping it does not have a right unit either. (G, ) is non-associative quasigroup. If θ(z) θ( z) z for some z G then (G, ) is non-commutative quasigroup.

13 The identity holds true in (G, ) iff θ l = I. The identity y = ((y x)... ) x } {{ } l x ( (x y)) = y } {{ } l holds true in (G, ) iff ( φ) l = I = (I θ) l.

14 If θ has a cycle containing 0 of length greater than G /2, then the quasigroup (G, ) cannot have a proper subquasigroup. The quasigroup (G, ) is without a proper subquasigroup if S G /2, where S = i=p+1 i=1 i=p+1 i=1 {θ i (0)} {λ i (0)} i=p+1 i=1 i=p+1 i=1 {θ(0) + θ i (0)} {λ(0) + λ i (0)} p = [ G /2] and λ = θ( φ) 1. i=p+1 i=1 i=p+1 i=1 {θ( θ i (0)) + θ i (0)} {λ( λ i (0)) + λ i (0)},

15 ((x x)... ) x = θ l (0) + x, } {{ } x ( (x x)) = ( φ) l (0) + x. } {{ } (5) l l ((x/ x)/... )/x = (θ 1 ) l (0) + x, } {{ } x/(... (x /x)) = (φθ 1 ) l (0) + x, } {{ } (6) l l ((x\x) \... ) \ x = ( θ( φ) 1 ) l (0)+x, x \ ( \ (x \x)) = (( φ) 1 ) l (0)+x, } {{ } } {{ } l l ((x// x)//... )//x } {{ } l ((x \\ x) \\... ) \\x } {{ } l (7) = ( φ( θ) 1 ) l (0)+x, x//(... //(x //x)) = ( ( θ) 1 ) l (0)+x, } {{ } l (8) = ( φ 1 ) l (0) + x, x \\( \\(x \\ x)) = (θφ 1 ) l (0) + x. } {{ } l (9)

16 (G, ) as an shapeless quasigroup Theorem Let θ be an orthomorphism of the abelian group (G, +), and let (G, ) be a quasigroup derived by θ by the equation (4). If θ is with the following properties: a) θ(0) 0 b) θ k I for all k < 2 G c) (I θ) k I for all k < 2 G d) θ(z) θ( Z) Z for some Z G e) S G /2, then (G, ) is a shapeless quasigroup.

17 Example of a shapeless quasigroup The abelian group (Z2 4, ) is given. x θ(x) φ(x) = x θ(x) We define the quasigroup operation as x y = θ(x y) y

18 Example of a shapeless quasigroup

19 1 2 3 Different generalizations of the Feistel Network as orthomorphisms 4 5

20 Different generalizations of the Feistel Network Feistel Networks are introduced by H. Feistel (Scientific American, 1973). Parameterized Feistel Network (PFN) (Markovski and Mileva (QRS 2009)), Extended Feistel networks type-1, type-2 and type-3 (Zheng et al (CRYPTO 1989)), Generalized Feistel-Non Linear Feedback Shift Register (GF-NLFSR) (Choy et al (ACISP 2009)). We will redefine the last two generalizations with parameters and over abelian groups.

21 Parameterized Feistel Network (PFN) Definition Let (G, +) be an abelian group, let f : G G be a mapping and let A, B, C G are constants. The Parameterized Feistel Network F A,B,C : G 2 G 2 created by f is defined for every l, r G by F A,B,C (l, r) = (r + A, l + B + f (r + C)).

22 Parameterized Feistel Network (PFN) In (Markovski and Mileva (QRS 2009)) is shown that if starting mapping f is bijection, the Parameterized Feistel Network F A,B,C and its square F 2 A,B,C are orthomorphisms of the group (G 2, +). More over, they are orthogonal orthomorphisms.

23 Parameterized Extended Feistel Network (PEFN) type-1 Definition Let (G, +) be an abelian group, let f : G G be a mapping, let A 1, A 2,..., A n+1 G are constants and let n > 1 be an integer. The Parameterized Extended Feistel Network (PEFN) type-1 F A1,A 2,...,A n+1 : G n G n created by f is defined for every (x 1, x 2,..., x n ) G n by F A1,A 2,...,A n+1 (x 1, x 2,..., x n ) = (x 2 + f (x 1 + A 1 ) + A 2, x 3 + A 3,..., x n + A n, x 1 + A n+1 ).

24 Parameterized Extended Feistel Network (PEFN) type-1

25 Parameterized Extended Feistel Network (PEFN) type-1 Theorem Let (G, +) be an abelian group, let n > 1 be an integer and A 1, A 2,..., A n+1 G. If F A1,A 2,...,A n+1 : G n G n is a PEFN type-1 created by a bijection f : G G, then F A1,A 2,...,A n+1 is an orthomorphism of the group (G n, +).

26 Parameterized Generalized Feistel-Non Linear Feedback Shift Register (PGF-NLFSR) Definition Let (G, +) be an abelian group, let f : G G be a mapping, let A 1, A 2,..., A n+1 G are constants and let n > 1 be an integer. The Parameterized Generalized Feistel-Non Linear Feedback Shift Register (PGF-NLFSR) F A1,A 2,...,A n+1 : G n G n created by f is defined for every (x 1, x 2,..., x n ) G n by F A1,A 2,...,A n+1 (x 1, x 2,..., x n ) = (x 2 + A 1, x 3 + A 2,..., x n + A n 1, x x n + A n + f (x 1 + A n+1 )).

27 Parameterized Generalized Feistel-Non Linear Feedback Shift Register (PGF-NLFSR)

28 Parameterized Generalized Feistel-Non Linear Feedback Shift Register (PGF-NLFSR) Theorem Let we use the abelian group (Z m 2, ), let n = 2k be an integer and A 1, A 2,..., A n+1 Z m 2. If F A 1,A 2,...,A n+1 : (Z m 2 )n (Z m 2 )n is a PGF-NLFSR created by a bijection f : Z m 2 Zm 2, then is an orthomorphism of the group ((Z m 2 )n, ). F A1,A 2,...,A n+1

29 1 2 3 Different generalizations of the Feistel Network as orthomorphisms 4 5

30 Quasigroups derived by the PFN F A,B,C Proposition Let (G, +) be an abelian group and let f : G G be a bijection. Let F A,B,C be a Parameterized Feistel Network created by f, and let (G 2, ) be a quasigroup derived by F A,B,C by the equation (4). If F A,B,C is with the following properties: a)a 0 or B f (C) b) F k A,B,C I for all k < 2 G 2 c) (I F A,B,C ) k I for all k < 2 G 2 d) S G 2 /2, then (G 2, ) is a shapeless quasigroup.

31 Quasigroups derived by the PFN F A,B,C We made a m-file in MatLab that produce a starting bijection f : Z2 n Z 2 n and parameters A, B and C for the orthomorphism PFN F A,B,C which produce a shapeless quasigroup. The group operation is XOR. For n = 3, 4, 5, 6 execution time is less than a minute and for n = 7 is less than five minutes. These results correspond to shapeless quasigroups of order 2 6, 2 8, 2 10, 2 12 and 2 14.

32 Quasigroups derived by the PEFN type-1 F A1,A 2,...,A n+1 Proposition Let (G, +) be an abelian group, let n > 1 be an integer, let A 1, A 2,..., A n+1 G and let f : G G be a bijection. Let F A1,A 2,...,A n+1 be a PEFN type-1 created by f, and let (G n, ) be a quasigroup derived by F A1,A 2,...,A n+1 by the equation (4). If F A1,A 2,...,A n+1 is with the following properties: a)a i 0 for some i {3, 4,..., n + 1} or A 2 f (A 1) b) FA k 1,A 2,...,A n+1 I for all k < 2 G n c) (I F A1,A 2,...,A n+1 ) k I for all k < 2 G n d) S G n /2, then (G n, ) is a shapeless quasigroup.

33 Quasigroups derived by the PGF-NLFSR F A1,A 2,...,A n+1 Proposition Let we use the abelian group (Z m 2, ), let n = 2k be an integer, let A 1, A 2,..., A n+1 Z m 2 and let f : Z m 2 Z m 2 be a bijection. Let F A1,A 2,...,A n+1 be a PGF-NLFSR created by f, and let ((Z m 2 ) n, ) be a quasigroup derived by F A1,A 2,...,A n+1 by the equation (4). If F A1,A 2,...,A n+1 is with the following properties: a) A i 0 for some i {1, 2,..., n 1} or A n f (A n+1) b) F k A 1,A 2,...,A n+1 I for all k < 2 G n c) (I F A1,A 2,...,A n+1 ) k I for all k < 2 G n d) S G n /2, then ((Z m 2 ) n, ) is a shapeless quasigroup.

34 1 2 3 Different generalizations of the Feistel Network as orthomorphisms 4 5

35 We give some constructions of shapeless quasigroups of different orders by using quasigroups produced by diagonal method from orthomorphisms. We show that PEFN type-1 produced by a bijection is an orthomorphism of the abelian group (G n, +) and that PGF-NLFSR produced by a bijection is an orthomorphism of the ((Z m 2 )n, ). Also, we parameterized these orthomorphisms for the need of cryptography, so we can work with different quasigroups in every iterations of the future cryptographic primitives.

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