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Polynomials and Finite Fields

Factoring Polynomials over Finite Fields

Factoring Polynomials over Finite Fields

... The Problem The Well-Known Methods The Main Idea Singular Hyperelliptic Curves Factoring f (x ). Factoring Polynomials over Finite Fields Enver Ozdemir[r] ...

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Factoring Polynomials Over Finite Fields: A Survey

Factoring Polynomials Over Finite Fields: A Survey

... Odlyzko, 1985; Lenstra, 1991), and for computing the number of points on elliptic curves (Buchmann, 1990). Major improvements have been made in the polynomial factorization problem during this decade both in theory and ...

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Factoring Multivariate Polynomials over Finite Fields

Factoring Multivariate Polynomials over Finite Fields

... corresponding fields go through with minor modifications when we are working with reducible algebraic sets and their corresponding ...prime fields while our algorithm works over all finite ...

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Galois Groups of Polynomials and the Construction of Finite Fields

Galois Groups of Polynomials and the Construction of Finite Fields

... constructing finite fields with the aid of Galois groups of polynomials of small ...these polynomials, their base fields and their splitting fields ...of finite ...

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Factoring Dickson polynomials over finite fields

Factoring Dickson polynomials over finite fields

... over finite fields, in “Finite Fields and Applications,” LMS Lecture Note Series, ...“Dickson Polynomials,” Pit- man Monographs and Surveys in Pure and Applied ...

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Several Classes of Permutation Polynomials over Finite Fields

Several Classes of Permutation Polynomials over Finite Fields

... Permutation Polynomials and Applications to Coding Theory. Finite Fields and Their Ap- plications, 13, ...Permutation Polynomials over Finite ...on Finite Fields and Their ...

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Some Problems Concerning Polynomials over Finite Fields, or Algebraic Divertissements

Some Problems Concerning Polynomials over Finite Fields, or Algebraic Divertissements

... over finite fields. These are functions such that their finite difference in every nonzero direction is a permutation polynomial of the ...of polynomials satisfying these ...

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A  new  public  key  system  based  on  Polynomials  over  finite  fields  GF(2)

A new public key system based on Polynomials over finite fields GF(2)

... signatures, polynomials over finite fields 1 Introduction Public-key cryptography started in 1976 with publication of pioneering work of Diffie and Hell- man [1] called DH key exchange and in 1978 ...

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On permutation polynomials over finite fields

On permutation polynomials over finite fields

... consider the problem of characterizing permutation polynomials; that is, we seek conditions on the coefficients of a polynomial which are necessary and sufficient for it to represent a p[r] ...

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Factoring polynomials over finite fields

Factoring polynomials over finite fields

... Let p an odd prime and let F p = Z/pZ the (unique up to automorphism) field with p-elements. We want to classify finite fields up to automorphism. Question 1 Give a field with 4 elements. Is there any field ...

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On some permutation polynomials over finite fields

On some permutation polynomials over finite fields

... Amir Akbary: Department of Mathematics and Computer Science, University of Lethbridge, 4401 University Drive West, Lethbridge, AB, Canada T1K 3M4. E-mail address : akbary@cs.uleth.ca[r] ...

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Finding irreducible polynomials over finite fields

Finding irreducible polynomials over finite fields

... fouiid in polvnomiil linic b^ niivc snrdi Ί(ιρ (ΐ\ is n«w ( tsil) si cn to bi rompiitiblc in.. pol>iH)iiuil tmR llif[r] ...

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Codes from Polynomials over Finite Fields

Codes from Polynomials over Finite Fields

... University of California, Irvine Colorado State Colloquium.. February 8, 2021.[r] ...

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Factorization Algorithms for Polynomials over Finite Fields

Factorization Algorithms for Polynomials over Finite Fields

... 1 Introduction Any non constant polynomial over a field can be expressed as a product of ir- reducible polynomials. In finite fields, some algorithms work for the calculation of irreducible factors of a polynomial of ...

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Factoring polynomials over special finite fields version

Factoring polynomials over special finite fields version

... The first author thanks the Information Technology Research Centre, Province of Ontario, for sponsoring a visit to the University of Toronto in 1988, during which a first version of this paper was written. 1. Introduction ...

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SUBQUADRATIC-TIME FACTORING OF POLYNOMIALS OVER FINITE FIELDS

SUBQUADRATIC-TIME FACTORING OF POLYNOMIALS OVER FINITE FIELDS

... of polynomials of degree less than deg(φ α ), such that for any u ∈ V we have φ α,u = φ α / gcd(φ α , L ...such polynomials only q deg( φ α ) −δ are divisible by ψ j , so the probability that ψ j does not ...

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Efficient Indexing of Necklaces and Irreducible Polynomials over Finite Fields

Efficient Indexing of Necklaces and Irreducible Polynomials over Finite Fields

... This leads us to one of the indexing problems studied here: Fix an alphabet Σ and consider two strings x and y in Σ n to be equivalent if one is a rotation of the other, i. e., we can find strings x 1 ,x 2 such that x = ...

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Factoring and Testing Irreducibility of Sparse Polynomials over Small Finite Fields

Factoring and Testing Irreducibility of Sparse Polynomials over Small Finite Fields

... As we already mentioned, x 2 d + x is the product of all irreducible polynomials of degree dividing d. For example, x 2 3 + x = x(x + 1)(x 3 + x + 1)(x 3 + x 2 + 1) . Thus, a simple (but slow) algorithm to find a ...

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On  Exponential  Sums,  Nowton  identities   and  Dickson  Polynomials  over  Finite  Fields

On Exponential Sums, Nowton identities and Dickson Polynomials over Finite Fields

... Case (1) f 1 (x, y) is invariant under the action of every σ ξ , ξ ∈ F q , or Case (2) there are some orbits, each of the form f t 1 (x, y), · · · , f t i (x, y) such that the union of these orbits is invariant under the ...

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PUBLIC  KEY  CRYPTOGRAPHY  USING  PERMUTATION  P-POLYNOMIALS  OVER  FINITE  FIELDS

PUBLIC KEY CRYPTOGRAPHY USING PERMUTATION P-POLYNOMIALS OVER FINITE FIELDS

... 4 Public key Cryptosystem In this section we present our multivariate public key cryptosystem using results from the previous section. Our cryptosystem will work in any arbitrary finite field F q m , m = p k . But ...

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