Number Of Irreducible Polynomials Research Articles

Batch generating keyed strong S-Boxes with high nonlinearity using 2D hyper chaotic map

Cryptanalysis result of the S-Box construction in AES revealed that, (1) the number of irreducible polynomials can be extend to 30, (2) if we ignore the existence of fixed point(s) and reverse fixed point(s) in an S-Box, the affine transformation constant c can select from [01, FF], and (3) the S-Box in AES is fixed, which causes potential security risks to the AES algorithm. Based on the above analysis, first we constructed a non-degenerate 2D enhanced quadratic map (2D-EQM) with ergodicity and randomness, and then used it to generate affine transformation matrices and affine transformation constants which were applied to seed S-Boxes to generate a batch of keyed strong S-Boxes with high nonlinearity. Furthermore, we counted that of all the generated 4.0913 × 1022 S-Boxes, about 5.4414 × 1020 of them can still keep the nonlinearity of 112. Finally, we used six criteria to evaluate the performance of constructed S-Boxes. Security and statistical analysis demonstrated the effectiveness and practicability of the proposed S-Box batch generating algorithm.

Three proofs of an observation on irreducible polynomials over GF(2)

We present three proofs of an observation of Ahmadi on the number of irreducible polynomials over GF(2) with certain traces and cotraces, the most interesting of which uses an explicit natural bijection. We also present two proofs of a related observation.

The number of irreducible polynomials over finite fields with vanishing trace and reciprocal trace

We present the formula for the number of monic irreducible polynomials of degree n over the finite field \({\mathbb {F}}_q\) where the coefficients of \(x^{n-1}\) and x vanish for \(n\ge 3\). In particular, we give a relation between rational points of algebraic curves over finite fields and the number of elements \(a\in {\mathbb {F}}_{q^n}\) for which Trace\((a)=0\) and Trace\((a^{-1})=0\).

The Chebotarev density theorem for function fields — Incomplete intervals

We prove a Pólya-Vinogradov type variation of the Chebotarev density theorem for function fields over finite fields valid for “incomplete intervals” I⊂Fp, provided (p1/2log⁡p)/|I|=o(1). Applications include density results for irreducible trinomials in Fp[x], i.e. the number of irreducible polynomials in the set {f(x)=xd+a1x+a0∈Fp[x]}a0∈I0,a1∈I1 is ∼|I0|⋅|I1|/d provided |I0|>p1/2+ϵ, |I1|>pϵ, or |I1|>p1/2+ϵ, |I0|>pϵ, and similarly when xd is replaced by any monic degree d polynomial in Fp[x]. Under the above assumptions we can also determine the distribution of factorization types, and find it to be consistent with the distribution of cycle types of permutations in the symmetric group Sd.

On the number of irreducible polynomials of special kinds in finite fields

Let $\mathbb{F}_q$ be the finite field of order $q$ and $f(x)$ be an irreducible polynomial of degree $n$ over $\mathbb{F} _q$. For a positive divisor $n_1$ of $n$, define the $n_1$-traces of $f(x)$ to be $\mathrm{Tr}(\alpha; n_1) = \alpha+\alpha^q+\cdots+\alpha^{q^{n_1-1}}$ where $\alpha$'s are the roots of $f(x)$. Let $N_q^*(n; n_1)$ denote the number of monic irreducible polynomials of degree $n$ over $\mathbb{F} _q$ with nozero $n_1$-traces. Ruskey, Miers and Sawada have found the formula for $N_q^*(n; n)$. Based on the properties of linearized polynomials, we obtain the formula for $N_q^*(n; n_1)$ in the general case, including a new proof to the result by Ruskey, Miers and Sawada.

The number of irreducible polynomials with the first two prescribed coefficients over a finite field

The number of irreducible polynomials with the first two prescribed coefficients over a finite field

On the reducibility of some composite polynomials over finite fields

Let g(x) = x n + a n-1 x n-1 + . . . + a 0 be an irreducible polynomial over $${\mathbb{F}_q}$$ . Varshamov proved that for a = 1 the composite polynomial g(x p ?ax?b) is irreducible over $${\mathbb{F}_q}$$ if and only if $${{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(nb-a_{n-1})\neq 0}$$ . In this paper, we explicitly determine the factorization of the composite polynomial for the case a = 1 and $${{\rm Tr}_{\mathbb{F}_q/\mathbb{F}_p}(nb-a_{n-1})= 0}$$ and for the case a ? 0, 1. A recursive construction of irreducible polynomials basing on this composition and a construction with the form $${g(x^{r^kp}-x^{r^k})}$$ are also presented. Moreover, Cohen's method of composing irreducible polynomials and linear fractions are considered, and we show a large number of irreducible polynomials can be obtained from a given irreducible polynomial of degree n provided that gcd(n, q 3 ? q) = 1.

Factorization of a class of polynomials over finite fields

We study the factorization of polynomials of the form F r ( x ) = b x q r + 1 − a x q r + d x − c over the finite field F q . We show that these polynomials are closely related to a natural action of the projective linear group PGL ( 2 , q ) on non-linear irreducible polynomials over F q . Namely, irreducible factors of F r ( x ) are exactly those polynomials that are invariant under the action of some non-trivial element [ A ] ∈ PGL ( 2 , q ) . This connection enables us to enumerate irreducibles which are invariant under [ A ] . Since the class of polynomials F r ( x ) includes some interesting polynomials like x q r − x or x q r + 1 − 1 , our work generalizes well-known asymptotic results about the number of irreducible polynomials and the number of self-reciprocal irreducible polynomials over F q . At the same time, we generalize recent results about certain invariant polynomials over the binary field F 2 .

Number of irreducible polynomials and pairs of relatively prime polynomials in several variables over finite fields

We discuss several enumerative results for irreducible polynomials of a given degree and pairs of relatively prime polynomials of given degrees in several variables over finite fields. Two notions of degree, the total degree and the vector degree, are considered. We show that the number of irreducibles can be computed recursively by degree and that the number of relatively prime pairs can be expressed in terms of the number of irreducibles. We also obtain asymptotic formulas for the number of irreducibles and the number of relatively prime pairs. The asymptotic formulas for the number of irreducibles generalize and improve several previous results by Carlitz, Cohen and Bodin.

The divisibility modulo 24 of Kloosterman sums on [formula omitted], m even

In a recent work by Charpin, Helleseth, and Zinoviev Kloosterman sums K ( a ) over a finite field F 2 m were evaluated modulo 24 in the case m odd, and the number of those a giving the same value for K ( a ) modulo 24 was given. In this paper the same is done in the case m even. The key techniques used in this paper are different from those used in the aforementioned work. In particular, we exploit recent results on the number of irreducible polynomials with prescribed coefficients.

On Irreducibles in the Polynomial Semigroup

We consider the asymptotic behavior of the number of irreducible polynomials over a finite field in arithmetic progressions.

The Number of Irreducible Polynomials and Lyndon Words with Given Trace

The trace of a degree n polynomial f(x) over GF(q) is the coefficient of xn-1 . Carlitz [Proc. Amer. Math. Soc., 3 (1952), pp. 693--700] obtained an expression Iq(n,t) for the number of monic irreducible polynomials over GF(q) of degree n and trace t. Using a different approach, we derive a simple explicit expression for Iq(n,t). If t>0, Iq(n,t) = (\sum \mu(d) q^{n/d})/(qn)$, where the sum is over all divisors d of n which are relatively prime to q. This same approach is used to count Lq(n,t), the number of q-ary Lyndon words whose characters sum to t mod q. This number is given by Lq(n,t) = (\sum {\rm gcd}(d,q) \mu(d) q^{n/d})/(qn)$, where the sum is over all divisors d of n for which gcd(d,q)|t. Both results rely on a new form of Möbius inversion.

Quantum Symmetric Algebras

The ‘plus part’ U+ of a quantum group Uq(g) has been identified by M. Rosso with a subalgebra Gsym of an algebra G which is a quantized version of R. Ree's shuffle algebra. Rosso has shown that Gsym and G – and hence also Hopf algebras which are analogues of quantum groups – can be defined in a much wider context. In this paper we study one of Rosso's quantizations, which depends on a family of parameters tij. Gsym is determined by a family of matrices Ωα whose coefficients are polynomials in the tij. The determinants of the Ωα factorize into a number of irreducible polynomials, and our main Theorem 5.2a gives strong information on these factors. This can be regarded as a first step towards the (still very distant!) goal, the classification of the symmetric algebras Gsym which can be obtained by giving special values to the parameters tij.

On the number of irreducible polynomials with 0,1 coefficients

On the number of irreducible polynomials with 0,1 coefficients

The least complex parallel massey-omura multiplier and its LCA and VLSI designs

It is known that for m≥4 there is more than one irreducible polynomial which generates a GF(2m) represented in normal basis. Therefore, a parallel Massey-Omura multiplier in GF(2m) has more than one structure for m≥4. The number of these structures is equal to the number of irreducible polynomials which can generate that specific Galois field in a normal-basis representation. In the paper, as an illustrative example, the structure of GF(25) multipliers for different generator polynomials are compared. Based on this comparison the least complex structure for this multiplier is introduced. The implementation of this structure using LCA (logic cell array) and VLSI (very large scale integration) technologies is given. Their complexities and propagation delays are compared.

The number of irreducible polynomials of a given form over a finite field

The number of irreducible polynomials of a given form over a finite field