Reconstructing noisy polynomial evaluation in residue rings

Reconstructing noisy polynomial evaluation in residue rings

Let q > 1 be an integer and let a and b be elements of the residue ring ZZq of integers modulo q. We show how, when given a polynomial f ∈ ZZq[X] and approximations to v0, v1 ∈ ZZq such that v1 ≡ f(v0) mod q one can recover v0 and v1 efficiently. This result has direct applications to predicting the polynomial congruential generator: a sequence (vn) of pseudorandom numbers defined by the relati...

Gray Images of Constacyclic Codes over some Polynomial Residue Rings

Let  be the quotient ring    where  is the finite field of size   and  is a positive integer. A Gray map  of length  over  is a special map from  to ( . The Gray map   is said to be a ( )-Gray map if the image of any -constacyclic code over    is a -constacyclic code over the field   . In this paper we investigate the existence of   ( )-Gray maps over . In this direction, we find an equivalent ...

Computations in Residue Class Rings of (Parametric) Polynomial Ideals

The paper proposes a new construction for studying various properties of a polynomial with respect to an ideal generated by a finite basis of polynomials. These properties include, among others, (i) checking whether the polynomial is a zero divisor in the residue class ring defined by the associated ideal and (ii) checking whether the polynomial is invertible in the residue class ring defined b...

Polynomial Functions over Rings of Residue Classes of Integers

In this thesis we discuss how to find equivalent representations of polynomial functions over the ring of integers modulo a power of a prime. Specifically, we look for lower degree representations and representations with fewer variables for which important applications in electrical and computer engineering exist. We present several algorithms for finding these compact formulations. INDEX WORD...

Differential Polynomial Rings of Triangular Matrix Rings