Finite Field Of Characteristic Research Articles

The Complete Differential Spectrum of a Class of Power Permutations Over Odd Characteristic Finite Fields

The Complete Differential Spectrum of a Class of Power Permutations Over Odd Characteristic Finite Fields

Minimal linear codes derived from weakly regular bent and plateaued functions

The construction of (minimal) linear codes from functions has received much attention in the literature. In this paper, we derive several minimal codes from the sets of pre-images of weakly regular plateaued and bent functions over the odd characteristic finite fields. Based on the recent construction method, we obtain six-weight and seven-weight minimal codes from these functions. The Hamming weights in proposed codes are calculated from the character sums of the sets based on the Walsh spectrum of the employed functions, while the weight distributions of the codes are determined from both the sizes of the employed sets and the Walsh distributions of the employed functions.

Construction of minimal linear codes with few weights from weakly regular plateaued functions

The construction of (minimal) linear codes from functions over finite fields has been greatly studied in the literature, since determining the parameters of codes based on functions is rather easy due to the nice structures of functions. In this paper, we derive $3$-weight and $4$-weight linear codes from weakly regular plateaued functions in the recent construction method of linear codes over the odd characteristic finite fields. The Hamming weights and weight distributions for codes are, respectively, determined by using the Walsh transform values and Walsh distributions of the employed functions. We also derive projective $3$-weight punctured codes with good parameters from the constructed codes. These punctured codes may be almost optimal due to the Griesmer bound, and they can be employed to design association schemes. We lastly show that all constructed codes are minimal, which approves that they can be employed to design high democratic secret sharing schemes.

Automorphisms of quasi-projective surfaces over fields of finite characteristic

We prove that the group of automorphisms of any quasi-projective surface S in finite characteristic has the p-Jordan property.

Uniform encodings to elliptic curves and indistinguishable point representation

Many cryptographic protocols which are based on elliptic curves require to efficiently encode bit-strings into the points of a given elliptic curve such that the encoding function satisfies computability, regularity, and samplability, or generally admissibility. All the admissible encoding functions from the finite field $$\mathbb {F}_q$$ are restricted to the class of elliptic curves with a non-trivial l-torsion $$\mathbb {F}_q$$ -rational point, where $$l\in \{2,3\}$$ . Therefore, there is no admissible encoding function to the class of many cryptographically interesting elliptic curves of prime order. In this paper, we present an admissible 2:1 encoding function from the set $$\{0,1,\ldots , \frac{q-1}{2}\}$$ to the $$\mathbb {F}_q$$ -rational points of arbitrary elliptic curves. We also propose an injective encoding function to elliptic curves with a non-trivial $$\mathbb {F}_q$$ -rational point of order two, that acts the same as the Bernstein et al.’s injective encoding function. Conversely, occasionally we have to transmit points of a known curve through an insecure channel. Traditional methods for transferring points enable an adversary to recognize patterns in the transmitted data. Consequently, one finds valuable information to attack the cryptosystem using the network traffic. By the help of the inverse of the injective encoding functions, Bernstein et al. introduced an interesting solution to this problem, namely Elligator. In this paper, we present an indistinguishable elliptic curve point representation using our given encoding function, which unlike the previous well-known encoding functions is not injective but covers almost all of elliptic curves over odd characteristic finite fields. Indeed, since we proposed a 2:1 encoding function to elliptic curves in short Weierstrass form, we have to select one pre-image randomly and transmit its corresponding bit-string instead of the point.

Differential Spectrum of Kasami Power Permutations Over Odd Characteristic Finite Fields

Functions with low differential uniformity have important applications in cryptography, coding theory, and sequence design. The differential spectrum of a cryptographic function is of great interest for estimating its resistance to some variants of differential cryptanalysis. Finding power permutations (i.e., monomial bijective mappings) over finite fields with low differential uniformity and determining their differential spectra have received a lot of attention over the past two decades. The objective of this paper is to study the differential properties of the well-known Kasami power permutations $x^{p^{2k}-p^{k}+1}$ over $ {\mathrm {GF}}(p^{n})$ , where $p$ is an odd prime and $k$ is an integer with $\gcd (n,k)=1$ . It turns out that this family of monomials is differentially $(p+1)$ -uniform. Our result in the case of $p=3$ gives an affirmative solution to a recent conjecture by Xu, Cao, and Xu. Most notably, the differential spectrum of this family of power permutations is completely determined.

Constructions of Codebooks Asymptotically Achieving the Welch Bound With Additive Characters

Codebooks with small inner product correlation have wide applications in many fields, such as direct spread code-division multiple access communications, signal processing, and compressed sensing. It is extremely hard to produce optimal codebooks with respect to the Welch bound or the Levenstein bound. This letter focuses on constructing asymptotically optimal codebooks with additive characters of finite fields. Following the proposed constructions, two classes of asymptotically optimal codebooks with respect to the Welch bound are obtained. In addition, the resulting codebooks could have a small alphabet size.

Deterministic construction of compressed sensing matrices with characters over finite fields

Compressed sensing theory provides a new approach to acquire data as a sampling technique and makes sure that an original sparse signal can be reconstructed from few measurements. The construction of compressed sensing matrices is a central problem in compressed sensing theory. In this paper, the deterministic compressed sensing matrices with characters of finite fields are constructed and the coherence of the matrices are computed. Furthermore, the maximum sparsity of recovering the original sparse signals by using our compressed sensing matrices is obtained. Meanwhile, a comparison is made with the compressed sensing matrices constructed by DeVore based on polynomials over finite fields. In the numerical simulations, our compressed sensing matrix outperforms DeVore’s matrix in the process of recovering original sparse signals.

Faster individual discrete logarithms in finite fields of composite extension degree

Computing discrete logarithms in finite fields is a main concern in cryptography. The best algorithms in large and medium characteristic fields (e.g., $\rm {GF}(p^2)$, $\rm {GF}(p^{12})$) are the Number Field Sieve and its variants (special, high-degree, tower). The best algorithms in small characteristic finite fields (e.g., $\rm {GF}(3^{6 \cdot 509})$) are the Function Field Sieve, Joux’s algorithm, and the quasipolynomial-time algorithm. The last step of this family of algorithms is the individual logarithm computation. It computes a smooth decomposition of a given target in two phases: an initial splitting, then a descent tree. While new improvements have been made to reduce the complexity of the dominating relation collection and linear algebra steps, resulting in a smaller factor basis (database of known logarithms of small elements), the last step remains at the same level of difficulty. Indeed, we have to find a smooth decomposition of a typically large element in the finite field. This work improves the initial splitting phase and applies to any nonprime finite field. It is very efficient when the extension degree is composite. It exploits the proper subfields, resulting in a much more smooth decomposition of the target. This leads to a new trade-off between the initial splitting step and the descent step in small characteristic. Moreover it reduces the width and the height of the subsequent descent tree.

Smoothness Test for Polynomials Defined Over Small Characteristic Finite Fields

The problem of determining whether a polynomial defined over a finite field ring is smooth or not with respect to a given degree, is the most intensive arithmetic operation of the so-called descent phase of index-calculus algorithms. In this paper, we present an analysis and efficient implementation of Coppersmith’s smoothness test for polynomials defined over finite fields with characteristic three. As a case study, we review the best strategies for obtaining a fast field and polynomial arithmetic for polynomials defined over the ring $$F_q[X],$$ with $$q=3^6,$$ and report the timings achieved by our library when computing the smoothness test applied to polynomials of several degrees defined in that ring. This software library was recently used in Adj et al. (Cryptology 2016. http://eprint.iacr.org/2016/914 ), as a building block for achieving a record computation of discrete logarithms over the 4841-bit field $${{\mathbb {F}}}_{3^{6\cdot 509}}$$ .

Having the same wild ramification is preserved by the direct image

Let S be the spectrum of an excellent henselian discrete valuation ring of residue characteristic p and X a separated scheme over S of finite type. Let $$\Lambda $$ and $$\Lambda '$$ be finite fields of characteristics $$\ell \ne p$$ and $$\ell '\ne p$$ respectively. For elements $$\mathcal {F}\in K_{c}(X,\Lambda )$$ and $$\mathcal {F}'\in K_{c}(X,\Lambda ')$$ of the Grothendieck groups of the categories of constructible sheaves of $$\Lambda $$ -modules and $$\Lambda '$$ -modules on X respectively, we introduce the notion that $$\mathcal {F}$$ and $$\mathcal {F}'$$ have the same wild ramification and prove that this condition is preserved by four of Grothendieck’s six operations except the derived tensor product and $$R \mathcal {H}om$$ .

Lq Norms of Fekete and Related Polynomials

AbstractA Littlewood polynomial is a polynomial in ℂ[z] having all of its coefficients in {−1, 1}. There are various old unsolved problems, mostly due to Littlewood and Erdos, that ask for Littlewood polynomials that provide a good approximation to a function that is constant on the complex unit circle, and in particular have small Lq normon the complex unit circle. We consider the Fekete polynomialswhere p is an odd prime and (· |p) is the Legendre symbol (so that z-1fp(z) is a Littlewood polynomial). We give explicit and recursive formulas for the limit of the ratio of Lq and L2 norm of fp when q is an even positive integer and p → ∞. To our knowledge, these are the first results that give these limiting values for specific sequences of nontrivial Littlewood polynomials and infinitely many q. Similar results are given for polynomials obtained by cyclically permuting the coefficients of Fekete polynomials and for Littlewood polynomials whose coefficients are obtained from additive characters of finite fields. These results vastly generalise earlier results on the L4 norm of these polynomials.

Technical history of discrete logarithms in small characteristic finite fields

Due to its use in cryptographic protocols such as the Diffie---Hellman key exchange, the discrete logarithm problem attracted a considerable amount of attention in the past 40 years. In this paper, we summarize the key technical ideas and their evolution for the case of discrete logarithms in small characteristic finite fields. This road leads from the original belief that this problem was hard enough for cryptographic purpose to the current state of the art where the algorithms are so efficient and practical that the problem can no longer be considered for cryptographic use.

Abelian varieties over fields of finite characteristic

Abstract The aim of this paper is to extend our previous results about Galois action on the torsion points of abelian varieties to the case of (finitely generated) fields of characteristic 2.

Correlation of binary sequence families derived from the multiplicative characters of finite fields

In this paper, new constructions of the binary sequence families of period $q-1$ with large family size and low correlation, derived from multiplicative characters of finite fields for odd prime powers, are proposed. For $m ≥ 2$, the maximum correlation magnitudes of new sequence families $\mathcal{S}_m$ are bounded by $(2m-2)\sqrt{q}+2m+2$, and the family sizes of $\mathcal{S}_m$ are given by $q-1$ for $m=2$, $2(q-1)-1$ for $m=3$, $(q^2-1)q^{\frac{m-4}{2}}$ for $m$ even, $m>2$, and $2(q-1)q^{\frac{m-3}{2}}$ for $m$ odd, $m>3$. It is shown that the known binary Sidel'nikov-based sequence families are equivalent to the new constructions for the case $m=2$.

Computing discrete logarithms in the Jacobian of high-genus hyperelliptic curves over even characteristic finite fields

We describe improved versions of index-calculus algorithms for solving discrete logarithm problems in Jacobians of high-genus hyperelliptic curves de ned over even characteristic elds. Our rst improvement is to incorporate several ideas for the low-genus case by Gaudry and Theriault, including the large prime variant and using a smaller factor base, into the large-genus algorithm of Enge and Gaudry. We extend the analysis in [24] to our new algorithm, allowing us to predict accurately the number of random walk steps required to nd all relations, and to select optimal degree bounds for the factor base. Our second improvement is the adaptation of sieving techniques from Flassenberg and Paulus, and Jacobson to our setting. The new algorithms are applied to concrete problem instances arising from the Weil descent attack methodology for solving the elliptic curve discrete logarithm problem, demonstrating signi cant improvements in practice.

The weight distributions of a class of cyclic codes II

Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ding et al. (IEEE Trans Inform Theory 57(12), 8000–8006, 2011); Ma et al. (IEEE Trans Inform Theory 57(1):397–402, 2011); Wang et al. (Trans Inf Theory 58(12):7253–7259, 2012); and Xiong (Finite Fields Appl 18(5):933–945, 2012). In this paper we use the method developed in Xiong (Finite Fields Appl 18(5):933–945, 2012) to solve one more special case. We make extensive use of standard tools in number theory such as characters of finite fields, the Gauss sums and the Jacobi sums. The problem of finding the weight distribution is transformed into a problem of evaluating certain character sums over finite fields, which turns out to be associated with counting the number of points on some elliptic curves over finite fields. We also treat the special case that the characteristic of the finite field is 2.

The weight distributions of a class of cyclic codes

Recently, the weight distributions of the duals of the cyclic codes with two zeros have been obtained for several cases in Ma et al. (2011) [14], Ding et al. (2011) [5], Wang et al. (2011) [20]. In this paper we provide a slightly different approach toward the general problem and use it to solve one more special case. We make extensive use of standard tools in number theory such as characters of finite fields, the Gauss sums and the Jacobi sums to transform the problem of finding the weight distribution into a problem of evaluating certain character sums over finite fields, which on the special case is related with counting the number of points on some elliptic curves over finite fields. Other cases are also possible by this method.

On isogeny classes of Edwards curves over finite fields

We count the number of isogeny classes of Edwards curves over odd characteristic finite fields, answering a question recently posed by Rezaeian and Shparlinski. We also show that each isogeny class contains a complete Edwards curve, and that an Edwards curve is isogenous to an original Edwards curve over F q if and only if its group order is divisible by 8 if q ≡ − 1 ( mod 4 ) , and 16 if q ≡ 1 ( mod 4 ) . Furthermore, we give formulae for the proportion of d ∈ F q ∖ { 0 , 1 } for which the Edwards curve E d is complete or original, relative to the total number of d in each isogeny class.

Efficient multiplications in [formula omitted] and [formula omitted

Efficient multiplications in finite fields of characteristics 5 and 7 are used for computing the Eta pairing over divisor class groups of the hyperelliptic curves Lee et al. (2008) [1]. In this paper, using the recent methods for multiplication in finite fields, the explicit formulas for multiplication in F 5 5 n and F 7 7 n are obtained with 10 multiplications in F 5 n for F 5 5 n and 15 multiplications in F 7 n for F 7 7 n improving the results in Cenk and Özbudak (2008) [4], Cenk et al. (2009) [5], Lee et al. (2008) [1] and Montgomery (2005) [12]. The timing results of implementations of the Karatsuba type formulas and the proposed formula for multiplication in F 5 5 ⋅ 89 are given.