Post on 23-Aug-2020
On the Multidimensional Distribution of NumbersGenerated 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, ep(x) = exp(2πIx/p) and Fp the finitefield with p elements,{0, . . . , p − 1}. X1, . . . ,Xk will denoteindeterminates and Fp[X1, . . . ,Xk ] will denote the ring ofpolynomials with coefficients in Fp over X1, . . . ,Xk .
Domingo Gomez University of Cantabria
On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials
Problem
Generate sequences (un) un ∈ Fkp , with good pseudorandom
properties.
I Good distribution properties;
I Difficult to predict;
I Easy to generate.
Domingo Gomez University of Cantabria
On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials
Pseudorandom Number Generators
Each element, un = (un,1, . . . , un,k) is defined by the followingrecurrence,
un+1,i = Fi (un,1, . . . , un,k), i = 1, . . . , k , n = 0, 1, . . .
For short, we will write,
un = F(un−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)i (X1, . . . ,Xk) = Fi (F
(n)1 (X1, . . . ,Xk), . . . ,F
(n)k (X1, . . . ,Xk)),
F (n) = {F (n)1 , . . . ,F
(n)k }.
Another way of defining the sequence (un) is,
un = F (n)(u0).
Domingo Gomez University of Cantabria
On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials
Results in the Multivariate Case
I NonLinear PRNG of Higher orders (Ostafe, Pelican,Shparlinski);
I Recursive PRNG based on Rational Functions (Ostafe,Shparlinski) generalizes Inversive Generator(Niederreiter,Rivat);
I Multivariate generalisation of the Power Generator (Ostafe,Shparlinski).
I 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 De(X , α)
De+2(X , α) = XDe+1(X , α)− αDe(X , α)
with
D0(X , α) = 2, D1(X , α) = X .
Domingo Gomez University of Cantabria
On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials
Properties of Dickson Polynomials
I De(X + αX−1, α) = X e + αeX−e .
I Df (De(X + αX−1, α), αe) = X ef + αef X−ef .
I De(x1, x21α) = xe
1De(1, α).
I If gcd(e, p2 − 1) = 1, then De(X , α) is a permutationalpolynomial.
Domingo Gomez University of Cantabria
On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials
Construction for k = 2
Take gcd(e, p2 − 1) = 1, define F = {F1(X1,X2),F2(X2))} where
F1(X1,X2) = De(X1,X2),
F2(X2) = X e2 .
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)1 (X1,X2) = Den(X1,X2),
F(n)2 (X2) = X en
2 .
Domingo Gomez University of Cantabria
On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials
Exponential Sums with This Generator
We start defining,
Sa1,a2(N) =N−1∑n=0
ep(a1un,1 + a2un,2)
and notice that∣∣∣∣∣Sa1,a2(N)−N−1∑n=0
ep(a1un+k,1 + a2un+k,2)
∣∣∣∣∣ ≤ 2k.
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)|Sa1,a2(N)| ≤W + (#K)K ,
where
W =N−1∑n=0
∣∣∣∣∣∑k∈K
ep (a1un+k,1 + a2un+k,2)
∣∣∣∣∣ .
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
W 2 ≤ NN−1∑n=0
∣∣∣∣∣∑k∈K
ep (a1un+k,1 + a2un+k,2)
∣∣∣∣∣2
≤ N∑k,`∈K
∑x1,x2∈Fp
ep(
a1(Dek (x1, x2)− De`(x1, x2))− a2(xek2 − xe`
2 ))
Domingo Gomez University of Cantabria
On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials
Exponential Sums with This Generator
To bound this sum∣∣∣∣∣∣∑
x1,x2∈Fp
ep(
a1(Dek (x1, x2)− De`(x1, x2))− a2(xek2 − xe`
2 ))∣∣∣∣∣∣ .
we notice that the following application,
x2 7→ x21x2,
is invertible if x1 6= 0. Making that substitution, we also notice that
Dek (x1, x2x21 ) = xek
1 Dek (1, x2), De`(x1, x2x21 ) = xe`
1 De`(1, x2).
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 ⊂ Zt , containing only units of Zt of cardinality K,any fixed δ > 0 and any integer h ≥ tδ there exists an integer rgcd(r , t) = 1, such that the number of solutions of the congruenceLr (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
t = p−1, K′ = {ek (mod p−1) | k = 0, 1, . . .}, h = p3/4T−1/2
where T is the multiplicative order of e modulo p − 1. Now, applythe transformation x1 7→ x r
1 here∑x1,x2∈Fp
ep(a1(x rek1 Drek (1, x2)− x re`
1 Dre`(1, x2))
− a2((x2r1 x2)e
k − (x2r1 x2)e
`))
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
t = p−1, K′ = {ek (mod p−1) | k = 0, 1, . . .}, h = p3/4T−1/2
where T is the multiplicative order of e modulo p − 1. Now, applythe transformation x1 7→ x r
1 here∑x1,x2∈Fp
ep(a1(xh11 Drek (1, x2)− xh2
1 Dre`(1, x2))
− a2(x2h11 (x2)e
k − x2h21 (x2)e
`))
Domingo Gomez University of Cantabria
On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials
Exponential Sums with This Generator
If k = ` then we use the trivial bound, otherwise, we use the Weilbound and this gives (#K)2hp3/2 + #Kp2. Substituting,
|Sa1,a2(N)| = O(N1/2T−1/4p9/8).
Domingo Gomez University of Cantabria
On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials
Remarks
I The previous bound is not trivial when N ≥ p;
I The multiplicative order of e modulo p − 1 must be large aswell;
I A bound for the exponential sum gives a bound for thediscrepancy using standard techniques. In this case,O(N−1/2T−1/4p9/8 log2(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 thatsimple monomials, with multipliers and shifts. Also the bounds forexponential 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 multivariateDickson polynomials. The construction is very similar butunfortunately there are some technical difficulties in the proof ofbounds 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
I Is there a way to improve the bound of the exponential sum?
I Is there a generalisation for pseudorandom number generatorswith Dickson polynomials for k > 2?
I Is it possible to add multipliers and still get a good lowerbound?
Domingo Gomez University of Cantabria
On the Multidimensional Distribution of Numbers Generated by Dickson Polynomials