Linearized Permutation Polynomials Research Articles

Cooperative Repair of Reed-Solomon Codes via Linearized Permutation Polynomials

Cooperative Repair of Reed-Solomon Codes via Linearized Permutation Polynomials

Linear permutations and their compositional inverses over 𝔽qn

The use of permutation polynomials over finite fields has appeared, along with their compositional inverses, as a good choice in the implementation of cryptographic systems. As a particular case, the construction of involutions is highly desired since their compositional inverses are themselves. In this work, we present an effective way of how to construct several linear permutation polynomials over [Formula: see text] as well as their compositional inverses using a decomposition of [Formula: see text] based on its primitive idempotents. As a consequence, involutions are also constructed.

New quadratic bent functions in polynomial forms with coefficients in extension fields

In this paper, we first discuss the bentness of a large class of quadratic Boolean functions in polynomial form $$f(x)=\sum _{i=1}^{{n}/{2}-1}\mathrm {Tr}^n_1(c_ix^{1+2^i})+ \mathrm {Tr}_1^{n/2}(c_{n/2}x^{1+2^{n/2}})$$ , where n is even, $$c_i\in \mathrm {GF}(2^n)$$ for $$1\le i \le {n}/{2}-1$$ and $$c_{n/2}\in \mathrm {GF}(2^{n/2})$$ . The bentness of these functions can be connected with linearized permutation polynomials. Hence, methods for constructing quadratic bent functions are given. Further, we consider a subclass of quadratic Boolean functions of the form $$f(x)=\sum _{i=1}^{{m}/{2}-1}\mathrm {Tr}^n_1(c_ix^{1+2^{ei}})+ \mathrm {Tr}_1^{n/2}(c_{m/2}x^{1+2^{n/2}})$$ , where $$n=em$$ , m is even, and $$c_i\in \mathrm {GF}(2^e)$$ . The bentness of these functions is characterized and some methods for deriving new quadratic bent functions are given. Finally, when m and e satisfy some conditions, we determine the number of these quadratic bent functions.

A simple method to determine the number of true different quadratic and cubic permutation polynomial based interleavers for turbo codes

Interleavers are important blocks of the turbo codes, their types and dimensions having a significant influence on the performances of the mentioned codes. If appropriately chosen, the permutation polynomial (PP) based interleavers lead to remarkable performances of these codes. The most used interleavers from this category are quadratic permutation polynomial (QPP) and cubic permutation polynomial (CPP) based ones. In this paper, we determine the number of different QPPs and CPPs that cannot be reduced to linear permutation polynomials (LPPs) or to QPPs or LPPs, respectively. They are named true QPPs and true CPPs, respectively. Our analysis is based on the necessary and sufficient conditions for the coefficients of second and third degree polynomials to be QPPs and CPPs, respectively, and on the Chinese remainder theorem. This is of particular interest when we need to find QPP or CPP based interleavers for turbo codes.

The limitation of permutation polynomial interleavers for turbo codes and a scheme for dithering permutation polynomials

In this letter, partial upper bounds on minimum distance for turbo codes with permutation polynomial (PP) based interleavers over integer rings are derived using the fact that PPs are equivalent to a family of linear permutation polynomials (LPPs). It is shown that upper bounds on minimum distance of turbo codes using higher order PP based interleavers are bounded by a function of the number of equivalent LPPs for PPs. Besides, it is shown that when the constant terms of LPPs are dithered, the resulting dithered LPP interleavers perform better than the quadratic permutation polynomial (QPP) based interleavers used in long term evolution (LTE) standard or than other good QPP or cubic permutation polynomial (CPP) based interleavers given in the literature.

The compositional inverse of a class of linearized permutation polynomials over [formula omitted], n odd

In this paper, the compositional inverses of a class of linearized permutation polynomials of the form P(x)=x+x2+tr(x/a) over the finite field F2n for an odd positive integer n, where “tr” is the trace function over F2n, are explicitly determined.

Linearized polynomials over finite fields revisited

We give new characterizations of the algebra Ln(Fqn) formed by all linearized polynomials reduced modulo (xqn−x) over the finite field Fqn after briefly surveying some known ones. One isomorphism we construct is between Ln(Fqn) and the composition algebra Fqn∨⊗FqFqn. The other isomorphism we construct is between Ln(Fqn) and the so-called Dickson matrix algebra Dn(Fqn). We also further study the relations between a linearized polynomial and its associate Dickson matrix, generalizing a well-known criterion of Dickson on linearized permutation polynomials. Adjugate polynomial of a linearized polynomial is then introduced, and connections between them are discussed. Both of the new characterizations can bring us new approaches to establish some special forms of representations of linearized polynomials proposed recently by several authors. Structure of the subalgebra Ln(Fqm) which is formed by all linearized polynomials reduced modulo (xqn−x) over a subfield Fqm of Fqn where m|n is also described.

Efficient Address Generation for Permutation Polynomial Based Interleavers over Integer Rings

Permutation polynomial based interleavers over integer rings have recently received attention for their excellent channel coding performance, elegant algebraic properties and simplicity of implementation. In this letter, it is shown that permutation polynomial based interleavers of practical interest is decomposed into linear permutation polynomials. Based on this observation, it is shown that permutation polynomial based interleavers as well as their inverses can be efficiently implemented.

A note on linearized polynomials and the dimension of their kernels

Recently explicit representations of the class of linearized permutation polynomials and the number of such polynomials were given in Zhou (2008) [4] and Yuan and Zeng (2011) [3]. In this paper, we generalize this result to linearized polynomials with kernel of any given dimension, solving an open problem in Charpin and Kyureghyan (2009) [1]. Moreover, more explicit representations of such polynomials are given and several classes of explicit linearized polynomials with kernel of any given dimension are presented.

New methods for generating permutation polynomials over finite fields

We present two methods for generating linearized permutation polynomials over an extension of a finite field F q . These polynomials are parameterized by an element of the extension field and are permutation polynomials for all nonzero values of the element. For the case of the extension degree being odd and the size of the ground field satisfying q ≡ 3 ( mod 4 ) , these parameterized linearized permutation polynomials can be used to derive non-parameterized nonlinear permutation polynomials via a recent result of Ding et al.

A note on linear permutation polynomials

Kai Zhou (2008) [8] gave an explicit representation of the class of linear permutation polynomials and computed the number of them. In this paper, we give a simple proof of the above results.

More explicit classes of permutation polynomials of [formula omitted

Permutation polynomials have been an interesting subject of study for a long time and have applications in many areas of science and engineering. However, only a small number of specific classes of permutation polynomials are known so far. In this paper, more linearized permutation polynomials and non-linearized permutation polynomials over F 3 3 m are presented.

A remark on linear permutation polynomials

We give an explicit representation of the class of linear permutation polynomials. By the representation, the number of them can be computed easily.

On maximal curves and linearized permutation polynomials over finite fields

The purpose of this paper is to construct maximal curves over large finite fields using linearized permutation polynomials. We also study linearized permutation polynomials under finite field extensions.

Linear permutation polynomials with coefficients in a subfield

Linear permutation polynomials with coefficients in a subfield