The Compositional Inverse of a Class of Permutation Polynomials Over a Finite Field∗

THE COMPOSITIONAL INVERSE OF A CLASS OF PERMUTATION POLYNOMIALS OVER A FINITE FIELD∗

ROBERT S. COULTER AND MARIE HENDERSON

Abstract. A new class of bilinear permutation polynomials was recently identiﬁed. In this note we determine the class of permutation polynomials which represents the functional inverse of the bilinear class.

1. Introduction and Main Result e Throughout Fq denotes the finite field of q = p elements for some prime p and positive integer e with Fq [X] representing the ring of polynomials in the indeterminate X over Fq . For polynomials f, g ∈ Fq [X], we write f ◦ g = f(g(X)) for the functional composition of f with g. A permutation polynomial over Fq is a polynomial which, under eval- uation, induces a permutation of the elements of Fq . Clearly, permutation polynomials are the only polynomials which have a (functional) inverse with respect to composition, id est for a permutation polyno- −1 mial f ∈ Fq [X] there exists (a unique) f ∈ Fq [X] of degree less than q such that f(f −1(X)) ≡ f −1(f(X)) ≡ X mod (Xq − X). We call f −1 the compositional inverse of f (or vice versa). The problem of discovering new classes of permutation polynomials is non-trivial and has generated much interest, see the surveys and open problems given in [3, 4, 6]. Discovering classes where the inverse polynomials can also be described seems to be even more difficult: there are very few known classes of permutation polynomials for which their compositional inverses are also known. To the authors knowledge, the classes with explicit formulae for inverses are:

(1) The linear polynomials: X + a where a ∈ Fq is trivially a permutation polynomial of Fq with the inverse polynomial being X − a.

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( + k and . . ) + T − X ( B + b 1 and X r( X − ( ) α 2 X . X B 1) ) ( b ) . X ( . ) α + )) X (iii), ( mo )) X α ) the . − = )) d 1 Bull. Austral. Math. Soc. 65 (2002), 521-526 So F b and The Prop Pr f Pr W it ( α or X e ( follo o o from g of. of then ) β α it result ( S X i ( = = = β osition =1 of ∈ k β i THE ws follo As ( X X β b c X k i F Equations The ha =2 T 1 α − ) X q ( )) directly 2 r( 1 T X X , i nk ws no v INVERSE =1 ) c k r( β from = e, or − α ) = w X β 1 T 5. + T em using β X + r( ( ) r( follo n f (2 b X j − X ) F α T X 2 Equation =1 k 1: that 1) ( ( or = r( − ) X g 2 ) / ws (2 OF 2 α Prop (2 2 β k β F and ) β k ( − β k X − or X T − i 2 f from A ) ∈ 2 r 2 2 ) β α )) k j ( k 2 x − osition k CLASS ( 2 − 3 X F k − i g k − i as ) ∈ q − i ) 1 α w 1 S ), ( S i ( calculating X then S + e i F i x required. ( it ( i X Q )) ha T X ( T X OF , is r( 4 ) r ) v = if ( f . ) + X and e simple β α PERMUT = = T x X ) ( c r( . g α + β X ) α i these T x Supp =1 k ( = X X r( ) i the β =1 k = to X β X 2 β ) 2 A ( T )) 2 ) k ose 0 sum iden X see TION 2 (2 T r( j then k = k r( 2 X − − nk T i tities, X 2 − β ) a − k r( 1 (2 POL − f 1 ( ) X x from 1 = α (2 k + ) − ( ) k S ) g 2 YNOMIALS − β T = 1 k α that + 2 2 − ( r( ( k k X i X − Prop 1 − ) b X S i i ( ) ) )) S for X i S ) for + ( i 2 . i X ( ). k ( X osition − c X x ) β α 1 . ) ) S ) ∈ . . ∈ k ( F F X Q q 4 5 ) , . Bull. Austral. Math. Soc. 65 (2002), 521-526 [6] [5] [4] [3] 6 Using [2] As elemen X f where using [1] The m fields. α utation ). ( G.L. R. R. W. A. and T graphs field?, the language cations amongst pro g w ec α determination e ( f Lidl, Lidl ceedings hnical, Blokh the Bosma, elements y Prop t α a β ha This Mullen, )) ( y survey II g v , ∈ and (D. p MA α and = ∈ , , e When G.L. uis, the ( olynomial osition Amer. J. 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