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On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials Domingo Gomez University of Cantabria Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Notation Let p be a prime number, e p ( x ) = exp(2 π Ix / p ) and F p the finite field with p elements, { 0 , . . . , p − 1 } . X 1 , . . . , X k will denote indeterminates and F p [ X 1 , . . . , X k ] will denote the ring of polynomials with coefficients in F p over X 1 , . . . , X k . Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Problem Generate sequences ( u n ) u n ∈ F k p , with good pseudorandom properties. ◮ Good distribution properties; ◮ Difficult to predict; ◮ Easy to generate. Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Pseudorandom Number Generators Each element, u n = ( u n , 1 , . . . , u n , k ) is defined by the following recurrence, u n +1 , i = F i ( u n , 1 , . . . , u n , k ) , i = 1 , . . . , k , n = 0 , 1 , . . . For short, we will write, u n = F ( u n − 1 ) Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Pseudorandom Number Generators Also, we introduce the following notation, F ( n +1) F i ( F ( n ) ( X 1 , . . . , X k ) , . . . , F ( n ) ( X 1 , . . . , X k ) = ( X 1 , . . . , X k )) , 1 i k { F ( n ) , . . . , F ( n ) F ( n ) = } . 1 k Another way of defining the sequence ( u n ) is, u n = F ( n ) ( u 0 ) . Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Results in the Multivariate Case ◮ NonLinear PRNG of Higher orders (Ostafe, Pelican, Shparlinski); ◮ Recursive PRNG based on Rational Functions (Ostafe, Shparlinski) generalizes Inversive Generator (Niederreiter,Rivat); ◮ Multivariate generalisation of the Power Generator (Ostafe, Shparlinski). ◮ Multivariate version of the Dickson Generator? Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Dickson Polynomials Denoted by D e ( X , α ) D e +2 ( X , α ) = XD e +1 ( X , α ) − α D e ( X , α ) with D 0 ( X , α ) = 2 , D 1 ( X , α ) = X . Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Properties of Dickson Polynomials ◮ D e ( X + α X − 1 , α ) = X e + α e X − e . ◮ D f ( D e ( X + α X − 1 , α ) , α e ) = X ef + α ef X − ef . ◮ D e ( x 1 , x 2 1 α ) = x e 1 D e (1 , α ) . ◮ If gcd( e , p 2 − 1) = 1, then D e ( X , α ) is a permutational polynomial. Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Construction for k = 2 Take gcd( e , p 2 − 1) = 1, define F = { F 1 ( X 1 , X 2 ) , F 2 ( X 2 )) } where F 1 ( X 1 , X 2 ) = D e ( X 1 , X 2 ) , X e F 2 ( X 2 ) = 2 . Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Construction for k = 2 Now, we notice that F ( n ) ( X 1 , X 2 ) = D e n ( X 1 , X 2 ) , 1 F ( n ) X e n ( X 2 ) = 2 . 2 Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Exponential Sums with This Generator We start defining, N − 1 � S a 1 , a 2 ( N ) = e p ( a 1 u n , 1 + a 2 u n , 2 ) n =0 and notice that � N − 1 � � � � � S a 1 , a 2 ( N ) − e p ( a 1 u n + k , 1 + a 2 u n + k , 2 ) � ≤ 2 k . � � � � n =0 Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Exponential Sums with This Generator For any set of integers K whose maximum is K ≥ 1, (# K ) | S a 1 , a 2 ( N ) | ≤ W + (# K ) K , where N − 1 � � � � � � W = e p ( a 1 u n + k , 1 + a 2 u n + k , 2 ) � . � � � � � n =0 k ∈K Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Exponential Sums with This Generator We use the Cauchy inequality to obtain 2 N − 1 � � W 2 ≤ N � � � � e p ( a 1 u n + k , 1 + a 2 u n + k , 2 ) � � � � � � n =0 k ∈K � � � � a 1 ( D e k ( x 1 , x 2 ) − D e ℓ ( x 1 , x 2 )) − a 2 ( x e k 2 − x e ℓ ≤ N 2 ) e p k ,ℓ ∈K x 1 , x 2 ∈ F p Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Exponential Sums with This Generator To bound this sum � � � � � � a 1 ( D e k ( x 1 , x 2 ) − D e ℓ ( x 1 , x 2 )) − a 2 ( x e k 2 − x e ℓ � � � 2 ) . e p � � � � x 1 , x 2 ∈ F p � � we notice that the following application, x 2 x 2 �→ 1 x 2 , is invertible if x 1 � = 0. Making that substitution, we also notice that 1 ) = x e k 1 ) = x e ℓ D e k ( x 1 , x 2 x 2 D e ℓ ( x 1 , x 2 x 2 1 D e k (1 , x 2 ) , 1 D e ℓ (1 , x 2 ) . Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

A Technical Lemma Lemma (Friedlander,Hansen,Shparlinski) For any set K ⊂ Z t , containing only units of Z t of cardinality K, any fixed δ > 0 and any integer h ≥ t δ there exists an integer r gcd( r , t ) = 1 , such that the number of solutions of the congruence L r ( h ) , rk ≡ y (mod t ) , k ∈ K , 0 ≤ y ≤ h − 1 , satisfies that is greater than certain constant times Kh / t. Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Exponential Sums with This Generator So, we apply last lemma with K ′ = { e k h = p 3 / 4 T − 1 / 2 t = p − 1 , (mod p − 1) | k = 0 , 1 , . . . } , where T is the multiplicative order of e modulo p − 1. Now, apply the transformation x 1 �→ x r 1 here e p ( a 1 ( x re k 1 D re k (1 , x 2 ) − x re ℓ � 1 D re ℓ (1 , x 2 )) x 1 , x 2 ∈ F p 1 x 2 ) e k − ( x 2 r 1 x 2 ) e ℓ )) − a 2 (( x 2 r Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Exponential Sums with This Generator So, we apply last lemma with K ′ = { e k h = p 3 / 4 T − 1 / 2 t = p − 1 , (mod p − 1) | k = 0 , 1 , . . . } , where T is the multiplicative order of e modulo p − 1. Now, apply the transformation x 1 �→ x r 1 here � e p ( a 1 ( x h 1 1 D re k (1 , x 2 ) − x h 2 1 D re ℓ (1 , x 2 )) x 1 , x 2 ∈ F p ( x 2 ) e k − x 2 h 2 ( x 2 ) e ℓ )) − a 2 ( x 2 h 1 1 1 Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Exponential Sums with This Generator If k = ℓ then we use the trivial bound, otherwise, we use the Weil bound and this gives (# K ) 2 hp 3 / 2 + # K p 2 . Substituting, | S a 1 , a 2 ( N ) | = O ( N 1 / 2 T − 1 / 4 p 9 / 8 ) . Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Remarks ◮ The previous bound is not trivial when N ≥ p ; ◮ The multiplicative order of e modulo p − 1 must be large as well; ◮ A bound for the exponential sum gives a bound for the discrepancy using standard techniques. In this case, O ( N − 1 / 2 T − 1 / 4 p 9 / 8 log 2 ( N )) . Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Further Remarks The multivariate power generator admits a more general form that simple monomials, with multipliers and shifts. Also the bounds for exponential sums are better than in this case. Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Even Further Remarks The generalisation for k ≥ 3 should be done using multivariate Dickson polynomials. The construction is very similar but unfortunately there are some technical difficulties in the proof of bounds for the corresponding exponential sums. Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials

Open Problems These are the open problems ◮ Is there a way to improve the bound of the exponential sum? ◮ Is there a generalisation for pseudorandom number generators with Dickson polynomials for k > 2? ◮ Is it possible to add multipliers and still get a good lower bound? Domingo Gomez University of Cantabria On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials