On some Constructions of Shapeless Quasigroups


 Percival Norman
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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 2527, 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 EDON80 (Gligoroski et al. (estream 2008)), hash functions EDONR (Gligoroski et al. (SHA )) and NaSHA (Markovski and Mileva (SHA )), digital signature algorithm MQQDSA (Gligoroski et al. (ACAM 2008)), public key cryptosystem LQLPs (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 nonidempotent, noncommutative, nonassociative, it does not have neither left nor right unit, it does not contain proper subquasigroups, 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 nonassociative quasigroup. If θ(z) θ( z) z for some z G then (G, ) is noncommutative 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 type1, type2 and type3 (Zheng et al (CRYPTO 1989)), Generalized FeistelNon Linear Feedback Shift Register (GFNLFSR) (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) type1 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) type1 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) type1
25 Parameterized Extended Feistel Network (PEFN) type1 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 type1 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 FeistelNon Linear Feedback Shift Register (PGFNLFSR) 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 FeistelNon Linear Feedback Shift Register (PGFNLFSR) 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 FeistelNon Linear Feedback Shift Register (PGFNLFSR)
28 Parameterized Generalized FeistelNon Linear Feedback Shift Register (PGFNLFSR) 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 PGFNLFSR 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 mfile 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 type1 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 type1 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 PGFNLFSR 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 PGFNLFSR 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 type1 produced by a bijection is an orthomorphism of the abelian group (G n, +) and that PGFNLFSR 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.
36 THANKS FOR YOUR ATTENTION!!!
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