Criterion of period maximality of trinomial over nontrivial Galois ring of odd characteristic

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1 Criterion of period maximality of trinomial over nontrivial Galois ring of odd characteristic V.N.Tsypyschev 1,, Ju.S.Vinogradova 1 a Russian State Social University, 4,W.Pik Str., Moscow, Russia Abstract In earlier eighties of XX century A.A.Nechaev has obtained the criterion of period fulfillment of a Galois polynomial over primarily residue ring Z 2 n. Also he has obtained necessary conditions of period maximality of the Galois polynomial over Z 2 n in the terminology of coefficients of its polynomial. Further A.S.Kuzmin has obtained analogous results for the case of Galois polynomial over primarily residue ring of odd characteristic. Later the first author of this article has carried the criterion of period fulfillment of the Galois polynomial over primarily residue ring of odd characteristic obtained by A.S.Kuzmin to the case of Galois polynomial over nontrivial Galois ring of odd characteristic. Using this criterion as a basis we have obtained criterion calling attention to. Keywords: Algebraic Shift Register, Galois ring, Polynomials of Maximal Period, Maximal Period Polynomial Synthesis 2010 MSC: 12, Introduction Let R = GR( n, p n ) be a Galois ring [7, 8], F (x) R[x] be a reversible unitary polynomial. Let T(F ) denote a period of polynomial F (x), i.e. minimal t with property: F (x) x λ (x t e) for some λ 0. Corresponding author address: tsypyschev@yandex.ru (V.N.Tsypyschev) Preprint submitted to Elsevier August 15, 2015

2 1 INTRODUCTION 2 Let F (x) be an image of F (x) under canonical epimorphism R[x] R[x]/pR[x]. Let remind [5], that : T( F (x)) T(F (x)) T( F (x)) p n 1. Polynomial F (x) is called distinguished, if T(F ) = T( F ), and is called full period polynomial, if T(F ) = T( F ) p n 1. Under additional condition T( F ) = m 1, polynomial F (x) is called maximal period polynomial (MP-polynomial). Unitary and reversible polynomial we call regular. Galois ring R = GR( n, p n ) is called nontrivial iff n > 1, p 2, i.e. iff R is neither field nor residue ring of integers. It is also well-known that the arbitrary element s R may be uniuely represented in the form s = n 1 i=0 γ i (s)p i, γ i (s) Γ(R), i = 0, n 1, (1.1) where Γ(R) = {x R x = x} is a p-adic coordinate set of the ring R (Teichmueller s representatives system). The set Γ(R) with operations : x y = (x + y) n and : x y = xy is a Galois field GF (). It is well-known [2] that for the synthesis of algebraic shift register over finite fields, rings and modules are necessary polynomials with high periodical ualities. A.A.Nechaev had obtained easily to verify necessary conditions of period maximality for polynomial over Z 2 n in terms of coefficients of this polynomial [6]. A.S.Kuzmin has carried this research to the case of primarily residue ring of integers Z p n, p 3 [6]. What follows is only application of results of [9] for obtaining of fulfillment period criterion for trinomials over nontrivial Galois ring R. Previously this result was published in Russian in the thesis form [10].

3 2 CRITERION OF TRINOMIALS PERIOD MAXIMALITY 3 Let s remind [6, 4, 9] that F (x) is a MP-polynomial over nontrivial Galois ring R of odd characteristic iff an image F (x) modulo p 2 R[x] is a MPpolynomial over Galois ring R = R/p 2 R = GR( 2, p 2 ). For convenience of referring let s remind also that Statement 1.1 ( [9]). Let F (x) R[x], R = GR( n, p n ), n > 1, p 3, p 2. If polynomial F (x) is represented in the form 1 F (x) = x t a t (x ), (1.2) t=0 for a t (x) = u t s=0 a is(t)x is(t), t = 0, 1, then regular Galois polynomial F (x) is a full period polynomial if and only if this condition holds: 1 F (x ) x t t=0 (c 0,t,...,c ut,t) Ω(t) ut s=0 c s,t! u t s=0 where family Ω(t), t = 0, 1 is defined as Ω(t) = { (c 0,t,..., c ut,t) c s,t 0, p, s = 0, u t, 2. Criterion of trinomials period maximality ( ais(t)x is(t)) c s,tp r 1 ( mod p 2 R ), } u t c s,t = p. s=0 (1.3) Theorem 2.1. Let G(x) be a polynomial over Galois ring R = GR( n, p n ), = p r, r 2, p 3, n 2, of the form G(x) = x m +ax k +b, and let T(Ḡ) = m 1. Then G(x) is a MP-polynomial over R if and only if at least one of following conditions hold: (I) m (II) m k, k 0; (III) m (IV) m 0, k 0, k 0, and additionally γ 1 (a) 0 or γ 1 (b) 0; 0; 0. Proof. To investigate period maximality of polynomial G(x) let s applicate relation (1.3). Let s consider residue classes modulo of degrees of non-zero monomials of polynomial G(x).

4 2 CRITERION OF TRINOMIALS PERIOD MAXIMALITY 4 These cases are possible: (a) m k 0. (b) m k, k 0. (c) m (d) m 0, k (e) m 0, k 0. The case (a). Because Ḡ(x) is a MP-polynomial then according to [1], numbers m è k are co-prime. Thus under conditions of the Theorem the case (a) is impossible. The case (b). Let m = t+i, k = j. Then right-hand side of the relation (1.3) takes a form x t x 2i + c 0,c 1 0,p: = x m + a x k + b + c 0!c 1! bc 0p r 1 a c 1p r 1 x jc 1p r 1 = c 0,c 1 1,p 1: c 0!c 1! bc 0p r 1 a c 1p r 1 x kc 1p r 1. For any c 1, c 1 1, p 1, c 1 c 1 these relations take place: and kc 1 p r 1 kc 1p r 1 0 < kc 1 p r 1, kc 1p r 1 < k < m. (2.1) Hence all members of the sum (2.1) are non-zero modulo p 2 R and has different degrees. It follows that the relation (1.3) takes place, i.e. all polynomials G(x) which satisfies conditions of Theorem and case (b) are MP-polynomyals. The case (c). The right-hand side of the relation (1.3) has a form: x m + a x k + b x m + γ 0 (a)x k + γ 0 (b) (mod p 2 R[x]). Hence the relation (1.3) takes place if and only if γ 1 (a) 0 or γ 1 (b) 0. The case (d). Let m = t + i, k = t + j. Then right-hand side of the relation (1.3) is eual b + x t c 0,c 1 0,p: c 0!c 1! ac 0p r 1 x c 0p r 1 j x c 1p r 1 i = = b + a x k + x m + x t c 0,c 1 1,p 1: c 0!c 1! ac 0p r 1 x pr 1 (c 0 j+c 1 i). (2.2)

5 2 CRITERION OF TRINOMIALS PERIOD MAXIMALITY 5 All members of the sum (2.2) are non=zero modulo p 2 R. Besides that, Hence t = c 0 p r 1 t + c 1 p r 1 t. t + c 0 p r 1 j + c 1 p r 1 i = c 0 p r 1 k + c 1 p r 1 m. It follows that under conditions c 0, c 0 1, p 1, c 0 c 0 and c 1 = p c 0, c 1 = p c 0 these euivalencies hold: t + c 0 p r 1 j + c 1 p r 1 i = t + c 0p r 1 j + c 1p r 1 i c 0 p r 1 k + c 1 p r 1 m = c 0p r 1 k + c 1p r 1 m c 0 k + c 1 m = c 0k + c 1m k(c 0 c 0) = m(c 1 c 1 ) k = m. Last euality is impossible. Hence members of the sum p 1 x t acpr 1 x pr 1 (cj+(p c)i) c=1 c!(p c)! (2.3) are of different degrees strictly less then m. Besides that summand a x k in the right-hand side of the euality (2.2) may allocate to zero not more than one of p 1 members of the sum (2.3). Because p 3 it means that relation (1.3) holds. So all polynomials G(x) which satisfies to conditions of the Theorem and the case (d) are of maximal period. The case (e). According to conditions of the Theorem, m = i, k = t+j. So the right-hand side of the relation (1.3) is eual to c 0,c 1 0,p: c 0!c 1! bc 0p r 1 x c 1p r 1 i + a x k = = b + a x k + x m + c 0,c 1 1,p 1: c 0!c 1! bc 0p r 1 x c 1p r 1m. (2.4) All members of the sum (2.4) are non-zero modulo p 2 R. For all c 1, c 1 1, p 1 such that c 1 c 1 these relations take place: c 1 p r 1 m c 1p r 1 m and c 1 p r 1 m < m.

6 3 CONCLUSION 6 Besides that the summand a x k in the right-hand side of the euality (2.4) may allocate to zero no more than one of p 1 others members of the sum p 1 c=1 c!(p c)! bcpr 1 x (p c)pr 1m. So because p 3 the relation (1.3) takes place. Hence all polynomials G(x) which satisfies to conditions of the Theorem and the case (e) has a maximal period. Theorem 2.2. Let G(x) be a polynomial over Galois ring R = GR( n, p n ), = p r, r 2, p 3, n 2, such that G(x) x m + ax k + b (mod p 2 R[x]), and T(Ḡ) = m 1. Then polynomial G(x) is a MP-polynomial over R if polynomial G(x) (mod p 2 R[x]) over p 2 R satisfies to conditions of the Theorem Conclusion So Theorem 2.2 provides us by method of easy verifying whenever a polynomial of special form has a maximal period. From other side the same Theorem provides an easy way of construction polynomials of maximal period. 4. Acknowledgments This work was supported by Russian State University for Humanities. Authors are very grateful to A.S.Kuzmin, A.A.Tarasov, S.Th.Petrov for a paid attention to this work and provided maintenance. [1] Albert A.A. Finite fields // Cybernetic summary ( a new series ) pp [2] Goresky, Mark; Klapper, Andrew // Algebraic shift register seuences, Cambridge: Cambridge University Press (ISBN /hbk). xv, 498 p., [3] Kuzmin A.S., Kurakin V.L., Mikhalev A.V., Nechaev A.A. Linear recurrences over rings and modules. // J. Math. Science (Contemporary Math. and its Appl., Thematic surveys) 1995 v.76 N6 p

7 4 ACKNOWLEDGMENTS 7 [4] Nechaev, A.A. Linear recurrence seuences over commutative rings. (English; Russian original) Discrete Math. Appl. 2, No.6, (1992); translation from Diskretn. Mat. 3, No.4, (1991).Zbl [5] Nechaev A.A. Cyclic typesof linear permutations over finite comutative rings // Matemat. sbornik 1993 ò.184 N4 C (in Russian) [6] Kuzmin A.S., Nechaev A.A. Linear recurrent seuences over Galois rings // II Int.Conf.Dedic.Mem. A.L.Shirshov Barnaul Aug (Contemporary Math. v p ) [7] McDonald C. Finite rings with identity // New York: Marcel Dekker p. [8] Radghavendran R. A class of finite rings // Compositio Math v.22 N1 p [9] Tsypyschev, V.N. Full periodicity of Galois polynomials over nontrivial Galois rings of odd characteristic. (English. Russian original) Zbl // J. Math. Sci., New York 131, No. 6, (2005); translation from Sovrem. Mat. Prilozh. 2004, No. 14, (2004) [10] Tsypyschev V.N., Vinogradova Ju.S. Criterion of period maximality of trinomial over nontrivial Galois ring of odd characteristic // Russian State University for Humanities bulletin Record management and Archival science, Informatics, Data Protection and Information Security series 18 pp (in Russian)

Criterion of maximal period of a trinomial over nontrivial Galois ring of odd characteristic

Criterion of maximal period of a trinomial over nontrivial Galois ring of odd characteristic V.N.Tsypyschev and Ju.S.Vinogradova Russian State Social University, 4,W.Pik Str., Moscow, Russia Abstract In