Finite Fields and Their Applications最新文献

A complete complementary code (CCC) consists of M sequence sets with size M. The sum of the auto-correlation functions of each sequence set is an impulse function, and the sum of cross-correlation functions of the different sequence sets is equal to zero. Thanks to their excellent correlation, CCCs received extensive use in engineering. In addition, they are strongly connected to orthogonal matrices. In some application scenarios, additional requirements are made for CCCs, such as recently proposed for concatenative CCC (CCCC) division multiple access (CCC-CDMA) technologies. In fact, CCCCs are a special kind of CCCs which requires that each sequence set in CCC be concatenated to form a zero-correlation-zone (ZCZ) sequence set. However, this requirement is challenging, and the literature is thin since there is only one construction in this context. We propose to go beyond the literature through this contribution to reduce the gap between their interest and our limited knowledge of CCCCs. This paper will employ novel methods for designing CCCCs and precisely derive two constructions of these objects. The first is based on perfect cross Z-complementary pair and Hadamard matrices, and the second relies on extended Boolean functions. Specifically, we highlight that optimal and asymptotic optimal CCCCs could be obtained through the proposed constructions. Besides, we shall present a comparison analysis with former structures in the literature and examples to illustrate our main results.

The classification of the 2-designs with $\lambda =2$ admitting a flag-transitive automorphism groups with socle $PSL(2,q)$ is completed by settling the two open cases in [2]. The result is achieved by using conics and hyperovals of $PG(2,q)$.

In this paper, we generalize [6], [1], [5] and [3] by allowing the distance between two points in a finite field vector space to be defined by a general non-degenerate bilinear form or quadratic form. We prove the same bounds on the sizes of large subsets of $F_q^d$ for them to contain distance graphs with a given maximal vertex degree, under the more general notion of distance. We also prove the same results for embedding paths, trees and cycles in the general setting.

In 2015, Abatzoglou, Silverberg, Sutherland, and Wong presented a framework for primality proving algorithms for special sequences of integers using an elliptic curve with complex multiplication. They applied their framework to obtain algorithms for elliptic curves with complex multiplication by imaginary quadratic field of class numbers one and two, but, they were not able to obtain primality proving algorithms in cases of higher class number. In this paper, we present a method to apply their framework to imaginary quadratic fields of class number three. In particular, our method provides a more efficient primality proving algorithm for special sequences of integers than the existing algorithms by using an imaginary quadratic field of class number three in which 2 splits. As an application, we give two special sequences of integers derived from $Q\left(\sqrt{-23}\right)$ and $Q\left(\sqrt{-31}\right)$, which are all the imaginary quadratic fields of class number three in which 2 splits. Finally, we give a computational result for the primality of these sequences.

Let q be a prime power. For $m=2,3,4$, we construct stable polynomials of the form $b^{m-1}{(x+a)}^m+c(x+a)+d$ over $F_q$ by Capelli's lemma. Moreover, when $m=2$ and $q\equiv 1\left(\mathrm{mod}4\right)$, we improve a lower bound for the number of stable quadratic polynomials over $F_q$ due to Goméz-Pérez and Nicolás [4]. When $m=3$, we prove Ahmadi and Monsef-Shokri's conjecture [2] that $x^3+x^2+1$ is stable over $F_2$.

Let q be a power of a prime such that $q\equiv 1\left(\mathrm{mod}r\right)$. Let c be an r-th power residue over $F_q$. In this paper, we present a new r-th root formula which generalizes G.H. Cho et al.'s cube root algorithm, and which provides a refinement of Williams' Cipolla-Lehmer based procedure. Our algorithm which is based on the recurrence relations arising from irreducible polynomial $h\left(x\right)=x^r+{(-1)}^{r+1}(b+{(-1)}^{r}r)(x-1)$ with $b=c+{(-1)}^{r+1}r$ requires only $O(r^2\log q+r^4)$ multiplications for $r>1$. The multiplications for computation of the main exponentiation of our algorithm are half of that of the Williams' Cipolla-Lehmer type algorithms.

This paper defines a linear representation for nonlinear maps $F:F^n\to F^n$ where $F$ is a finite field, in terms of matrices over $F$. This linear representation of the map F associates a unique number N and a unique matrix M in $F^{N\times N}$, called the Linear Complexity and the Linear Representation of F respectively, and shows that the compositional powers $F^{\left(k\right)}$ are represented by matrix powers $M^k$. It is shown that for a permutation map F with representation M, the inverse map has the linear representation $M^{-1}$. This framework of representation is extended to a parameterized family of maps $F_{\lambda }\left(x\right):F\to F$, defined in terms of a parameter $\lambda \in F$, leading to the definition of an analogous linear complexity of the map $F_{\lambda }\left(x\right)$, and a parameter-dependent matrix representation $M_{\lambda }$ defined over the univariate polynomial ring $F\left[\lambda \right]$. Such a representation leads to the construction of a parametric inverse of such maps where the condition for invertibility is expressed through the unimodularity of this matrix representation $M_{\lambda }$. Apart from computing the compositional inverses of permutation polynomials, this linear representation is also used to compute the cycle structures of the permutation map. Lastly, this representation is extended to a representation of the cyclic group generated by a permutation map F, and to the group generated by a finite number of permutation maps over $F$.

Cryptographic group actions provide a flexible framework that allows the instantiation of several primitives, ranging from key exchange protocols to PRFs and digital signatures. The security of such constructions is based on the intractability of some computational problems. For example, given the group action $(G,X,\star )$, the weak unpredictability assumption (Alamati et al. (2020) [1]) requires that, given random $x_i$'s in X, no probabilistic polynomial time algorithm can compute, on input ${\left\{\right(x_i,g\star x_i\left)\right\}}_{i=1,\dots ,Q}$ and y, the set element $g\star y$.

In this work, we study such assumptions, aided by the definition of group action representations and a new metric, the q-linear dimension, that estimates the “linearity” of a group action, or in other words, how much it is far from being linear. We show that under some hypotheses on the group action representation, and if the q-linear dimension is polynomial in the security parameter, then the weak unpredictability and other related assumptions cannot hold. This technique is applied to some actions from cryptography, like the ones arising from the equivalence of linear codes, as a result, we obtain the impossibility of using such actions for the instantiation of certain primitives.

As an additional result, some bounds on the q-linear dimension are given for classical groups, such as $S_n$, $\mathrm{GL}\left(F^n\right)$ and the cyclic group $Z_n$ acting on itself.

We study the stopping time of the Collatz map for a polynomial $f\in F_2\left[x\right]$, and bound it by $O\left(\mathrm{deg}{\left(f\right)}^{1.5}\right)$, improving upon the quadratic bound proven by Hicks, Mullen, Yucas and Zavislak. We also prove the existence of arithmetic sequences of unbounded length in the stopping times of certain sequences of polynomials, a phenomenon observed in the classical Collatz map.

The study of n-dimensional inverted Kloosterman sums was suggested by Katz (1995) [7] who handled the case when $n=1$ from complex point of view. For general $n\ge 1$, the n-dimensional inverted Kloosterman sums were studied from both complex and p-adic point of view in our previous paper (2024) [10]. In this note, we study the algebraic degree of the inverted n-dimensional Kloosterman sum as an algebraic integer.