PPs over Finite Fields defined by functional equations

Transcription

1 Permutation Polynomials over Finite Fields defined by functional equations Neranga Fernando Joint work with Dr. Xiang-dong Hou Department of Mathematics and Statistics University of South Florida AMS Spring Southeastern Section Meeting, Tampa, Florida March 10,2012

2 Outline 1. Introduction 2. Desirable Triples 3. Families of Desirable Triples 4. An update of the table of Desirable Triples (n, e; q) with q = 3 and e 6

3 Introduction The Polynomial g n,q q = p k, n 0. There exists a unique polynomial g n,q F p [x] such that (x + a) n = g n,q (x q x) a F q Question : When is g n,q a permutation polynomial(pp) of F q e? If g n,q is a PP, we call triple (n, e; q) desirable.

4 Introduction Recurrence g 0,q =... = g q 2,q = 0, g q 1,q = 1, g n,q = xg n q,q + g n q+1,q, n q Above recurrence relation can be used to define g n,q for n < 0 : g n,q = 1 x (g n+q,q g n+1,q ). For n < 0, there exists a g n,q F p [x, x 1 ] such that (x + a) n = g n,q (x q x) a F q Recurrence relation holds for all n Z.

5 Introduction g n,q and Reversed Dickson Polynomial The nth Reversed Dickson Polynomial D n (1, x) Z[x] is defined by D n (1, x(1 x)) = x n + (1 x) n When q = 2, g n,2 (x(1 x)) = x n + (1 x) n F 2 [x] g n,2 = D n (1, x) F 2 [x].

6 Desirable Triples Equivalence 1. g pn,q = g p n,q.

7 Desirable Triples Equivalence 1. g pn,q = g p n,q. 2. If n 1, n 2 > 0 are integers such that n 1 n 2 (mod q pe - 1 ), then g n1,q g n2,q (mod x qe x).

8 Desirable Triples Equivalence 1. g pn,q = g p n,q. 2. If n 1, n 2 > 0 are integers such that n 1 n 2 (mod q pe - 1 ), then g n1,q g n2,q (mod x qe x). 3. If m, n > 0 belong to the same p-cyclotomic coset modulo q pe - 1, we say that two triples (m, e; q) and (n, e; q) are equivalent and write (m, e; q) (n, e; q). If (m, e; q) (n, e; q), g m,q is a PP if and only if g n,q is a PP.

9 Desirable Triples Some Necessary Conditions If (n, e; 2) is desirable, gcd(n, 2 2e 1) = 3. If (n, e; q) is desirable, gcd(n, q 1) = 1. If (n, e; q) is desirable where q > 2 or e > 1, the p-cyclotomic coset of n modulo q pe 1 has cardinality pek(q = p k ).

10 Generating function A Quick Reminder : g 0,q =... = g q 2,q = 0, g q 1,q = 1, g n,q = xg n q,q + g n q+1,q, n q g n,q t n = n 0 t q 1 1 t q 1 xt q

11 Theorem g n,q t n (xt) q 1 1 (xt) q 1 (xt) q + (1 x q 1 ) tq 1 1 t q 1 (mod x q x) n 0 x 1 : if n > 0, n 0 (mod q 1) (mod x q x) g n,q a n x n + 0 : otherwise (mod x q x) where a n t n t q 1 = 1 t q 1 t q n 0

12 Proof of the Theorem g n,q t n = n 0 t q 1 1 t q 1 xt q and t q 1 1 t q 1 xt q t q 1 1 t q 1 xt q (xt) q 1 1 (xt) q 1 (xt) q + (1 x q 1 ) tq 1 1 t q 1 (mod x q 1 1) (xt) q 1 1 (xt) q 1 (xt) q + (1 x q 1 ) tq 1 1 t q 1 (mod x)

13 The case e = 1 (n, 1; q) is desirable if and only if gcd(n, q 1) = 1 and a n 0. Proof : By the Theorem, we have x q 1 1 : if n > 0, n 0 (mod q 1) (mod x q x) g n,q a n x n + 0 : otherwise (mod x q x) ( ) Since g n,q is PP, by a previous fact, gcd(n, q 1) = 1. So g n,q (x) a n x n (mod x q x) which implies a n 0. ( ) gcd(n, q 1) = 1 and a n 0 g n,q (x) a n x n (mod x q x) g n,q is PP. Open question : Determine n s.t. a n 0.

14 Some Desirable Triples [Hou 2011] 1. (q pe 2, e; q) when q > 2.

15 Some Desirable Triples [Hou 2011] 1. (q pe 2, e; q) when q > When q = 3 k, (q 2e q e 1, e; q) is desirable.

16 Some Desirable Triples [Hou 2011] 1. (q pe 2, e; q) when q > When q = 3 k, (q 2e q e 1, e; q) is desirable. 3. (3 2e e 2, e; 3) is desirable.

17 A useful Lemma - (A) Let n = α 0 q α t q t base q weight of n,, 0 α i q 1 and w q (n) be the 0 : if w q(n) < q 1 g n,q = 1 : if w q(n) = q 1 α 0 x q0 + (α 0 + α 1 )x q (α α t 1 )x qt 1 + δ : if w q(n) = q where δ = { 1 : if q = 2 0 : if q > 2

18 Proposition Let n = α 0 q α t q t, 0 α i q 1, with w q (n) = q. Then (n, e; q) is desirable if and only if gcd(α 0 + (α 0 + α 1 )x (α α t 1 )x t 1, x e 1) = 1. We are going to write n = α(p 0e + p 1e p (p 1)e ) + β, where α, β Z and impose conditions on α and β so that (n, e; p) is desirable.

19 Another Useful Lemma - (B) Let n = α(p 0e + p 1e p (p 1)e ) + β, for x F p e, where α, β Z. Then g n,p (x) = { gαp+β,p (x) : if Tr Fp e /F p (x) = 0 x α g β,p (x) : if Tr Fp e /F p (x) 0

20 Proposition In the previous Lemma, (n, e; p) is desirable if the following two conditions are satisfied. (i) Both g αp+β,p + δ and x α g β,p are F p - linear on F p e and are 1 1 on Tr 1 F p e /F p (0) = {x F p e : Tr Fp e /F p (x) = 0}, (ii) g β,p (1) eδ

21 A Desirable Family - 1 [Hou 2011] Let n = 8( e + 3 2e ) + 7. (α = 8, β = 7) g n,3 = { g8 3+7,3 (x) = g 31,3 (x) = x 30 x 31 x 32 : if Tr F3 e /F 3 (x) = 0 x 8 g 7,3 (x) = x 9 : if Tr F3 e /F 3 (x) 0 g n,3 is a PP of F 3 e if and only if g 31,3 is 1-1 on Tr 1 F 3 e /F 3 (0). g 31,3 (x 3 x) = x + x 3 + x 33. g 31,3 is 1-1 on Tr 1 F 3 e /F 3 (0) if and only if gcd(1 + x + x 3, x e 1) = x 1. (n, e; 3) is desirable if and only if gcd(1 + x + x 3, x e 1) = x 1.

22 A Desirable Family - 2 Let n = 26( e + 3 2e ) + 7. (α = 26, β = 7) g n,3 = g ,3 (x) = g 85,3 (x) = x 30 x 31 x 32 x 33 : if Tr F3 e /F 3 (x) = 0 x 26 g 7,3 (x) = x 27 : if Tr F3 e /F 3 (x) 0 g n,3 is a PP of F 3 e if and only if g 85,3 is 1-1 on Tr 1 F 3 e /F 3 (0). g 85,3 (x 3 x) = x 30 + x 31 + x 34. g 85,3 is 1-1 on Tr 1 F 3 e /F 3 (0) if and only if gcd(1 + x + x 4, x e 1) = x 1. (n, e; 3) is desirable if and only if gcd(1 + x + x 4, x e 1) = x 1.

23 A Desirable Family - 3 Let n = 163( e + 3 2e ) 162. (α = 163, β = 162) g n,3 = g ,3 (x) = g 327,3 (x) = x 31 + x 32 + x 33 x 34 : if Tr F3 e /F 3 (x) = 0 x 163 g 162,3 (x) = x : if Tr F3 e /F 3 (x) 0 g n,3 is a PP of F 3 e if and only if g 327,3 is 1-1 on Tr 1 F 3 e /F 3 (0). g 327,3 (x 3 x) = x 31 + x 34 + x 35. g 327,3 is 1-1 on Tr 1 F 3 e /F 3 (0) if and only if gcd(x + x 4 + x 5, x e 1) = x 1. (n, e; 3) is desirable if and only if gcd(x + x 4 + x 5, x e 1) = x 1.

24 A Desirable Family - 4 [Hou 2011] Let n = 4( e + 3 2e ) 7. (α = 4, β = 7) Then (n, e; 3) is desirable. g n,3 = x + x 1 x 3 { g4 3 7,3 (x) = g 5,3 (x) = x : if Tr F3 e /F 3 (x) = 0 x 4 g 7,3 (x) = x + x 1 x 3 : if Tr F3 e /F 3 (x) 0 is 1-1 on F 3 e \Tr 1 F 3 e /F 3 (0). ( Proved in the Paper Permutation Polynomials of the form (x p x + δ) s + L(x) by J. Yuan, C. Ding, H. Wang, J. Pieprzyk, 2008)

25 Table - [Hou 2011] - New Results Table : Desirable pairs (n, e; 3), e 6 e n 3-adic digits of n reference

26 Table Table : Desirable pairs (n, e; 3), e 6 e n 3-adic digits of n reference

27 Table Table : Desirable pairs (n, e; 3), e 6 e n 3-adic digits of n reference

28 Table Table : Desirable pairs (n, e; 3), e 6 e n 3-adic digits of n reference

29 Table Table : Desirable pairs (n, e; 3), e 6 e n 3-adic digits of n reference

30 Table Table : Desirable pairs (n, e; 3), e 6 e n 3-adic digits of n reference

31 Table Table : Desirable pairs (n, e; 3), e 6 e n 3-adic digits of n reference

32 Table Table : Desirable pairs (n, e; 3), e 6 e n 3-adic digits of n reference

33 Table Table : Desirable pairs (n, e; 3), e 6 e n 3-adic digits of n reference

34 Table Table : Desirable pairs (n, e; 3), e 6 e n 3-adic digits of n reference

35 References (1) X. H., A new Approach to Permutation Polynomials over finite fields, Finite Fields Appl. (2011). (2) X. H., G.L. Mullen, J.A. Sellers, J.L. Yucas, Reversed Dickson Polynomials over Finite Fields, Finite Fields Appl. 15 (2009), (3) X. H., Two classes of Permutation Polynomials over Finite Fields, J. Combin. Theory A, 118 (2011), (4) X. H. and T. Ly, Necessary conditions for reversed Dickson polynomials to be permutational, Finite Fields Appl. 16 (2010) (5) J. Yuan, C. Ding, H. Wang, J. Pieprzyk, Permutation Polynomials of the form (x p x + δ) s + L(x), Finite Fields Appl. 14 (2008),

36 Thank You!