Prime Field Research Articles

The squaring operation and the hit problem for the polynomial algebra in a type of generic degree

Let Pk be the graded polynomial algebra F2[x1,x2,…,xk] with the degree of each generator xi being 1, where F2 denote the prime field with two elements.The hit problem of Frank Peterson asks for a minimal generating set for the polynomial algebra Pk as a module over the mod-2 Steenrod algebra A. Equivalently, we want to find a vector space basis for F2⊗APk in each degree.In this paper, we study a generating set for the kernel of Kameko's squaring operation Sq˜⁎0:F2⊗APk⟶F2⊗APk in a so-called generic degree. By using this result, we explicitly compute the hit problem for k=5 in the respective generic degree.

On higher-order Fourier analysis in characteristic p

AbstractIn this paper, the nilspace approach to higher-order Fourier analysis is developed in the setting of vector spaces over a prime field $\mathbb {F}_p$ , with applications mainly in ergodic theory. A key requisite for this development is to identify a class of nilspaces adequate for this setting. We introduce such a class, whose members we call p-homogeneous nilspaces. One of our main results characterizes these objects in terms of a simple algebraic property. We then prove various further results on these nilspaces, leading to a structure theorem describing every finite p-homogeneous nilspace as the image, under a nilspace fibration, of a member of a simple family of filtered finite abelian p-groups. The applications include a description of the Host–Kra factors of ergodic $\mathbb {F}_p^\omega $ -systems as p-homogeneous nilspace systems. This enables the analysis of these factors to be reduced to the study of such nilspace systems, with central questions on the factors thus becoming purely algebraic problems on finite nilspaces. We illustrate this approach by proving that for $k\leq p+1$ the kth Host–Kra factor is an Abramov system of order at most k, extending a result of Bergelson–Tao–Ziegler that holds for $k< p$ . We illustrate the utility of p-homogeneous nilspaces also by showing that the structure theorem yields a new proof of the Tao–Ziegler inverse theorem for Gowers norms on $\mathbb {F}_p^n$ .

Noninjectivity of the cycle class map in continuous -adic cohomology

AbstractJannsen asked whether the rational cycle class map in continuous$\ell $-adic cohomology is injective, in every codimension for all smooth projective varieties over a field of finite type over the prime field. As recently pointed out by Schreieder, the integral version of Jannsen’s question is also of interest. We exhibit several examples showing that the answer to the integral version is negative in general. Our examples also have consequences for the coniveau filtration on Chow groups and the transcendental Abel-Jacobi map constructed by Schreieder.

The Structure of Reed–Muller Codes Over a Nonprime Field

It is well known that Reed–Muller codes over a prime field are radical powers of a corresponding group algebra. The case of a nonprime field is less studied in terms of equalities and inclusions between Reed–Muller codes and radical powers. In this paper, we prove that Reed–Muller codes in the case of a nonprime field of arbitrary characteristic are distinct from radical powers and provide necessary and sufficient conditions for inclusions between these codes and the powers of the radical.

The Huff curve–ElGamal graphic public key cryptosystem

A new graph for the cryptographic application is presented in this work. This graph is defined using the Huff curve over a prime field. The Huffs curve graph (HCg) considered as an alternative graph of the Edwards curve graph (EdCG).The HCg is used to give another modified Version Public-key cryptosystem ElGamal (EPKC) the proposed HC-ElGamal graphic cryptosystem enhances the secure communication in compare with original EPKC.

Constructing Representative Collective Signature Protocols Using The GOST R34.10-1994 Standard

The representative collective digital signature, which was suggested by us, is built based on combining the advantages of group digital signature and collective digital signature. This collective digital signature schema helps to create a unique digital signature that deputizes a collective of people representing different groups of signers and may also include personal signers. The advantage of the proposed collective signature is that it can be built based on most of the well-known difficult problems such as the factor analysis, the discrete logarithm and finding modulo roots of large prime numbers and the current digital signature standards of the United States and Russian Federation. In this paper, we use the discrete logarithmic problem on prime finite fields, which has been implemented in the GOST R34.10-1994 digital signature standard, to build the proposed collective signature protocols. These protocols help to create collective signatures: Guaranteed internal integrity and fixed size, independent of the number of members involved in forming the signature. The signature built in this study, consisting of 3 components (U, R, S), stores the information of all relevant signers in the U components, thus tracking the signer and against the “disclaim of liability” of the signer later is possible. The idea of hiding the signer’s public key is also applied in the proposed protocols. This makes it easy for the signing group representative to specify which members are authorized to participate in the signature creation process.

Inverse theorem for certain directional Gowers uniformity norms

Let G be a finite-dimensional vector space over a prime field Fp with some subspaces H1,...,Hk. Let f : G ? C be a function. Generalizing the notion of Gowers uniformity norms, Austin introduced directional Gowers uniformity norms of f over (H1,...,Hk) as ||f||2kU(H1,...,Hk) = E x?G,h1?H1,...,hk?Hk ? ?h1... ??hk f(x) where ??uf(x) : = f(x + u)?f(x) is the discrete multiplicative derivative. Suppose that G is a direct sum of subspaces G = U1 ? U2 ?...? Uk. In this paper we prove the inverse theorem for the norm ||?||U(U1,...,Uk,G,...,G), with ? copies of G in the subscript, which is the simplest interesting unknown case of the inverse problem for the directional Gowers uniformity norms. Namely, writing ||? ||U for the norm above, we show that if f : G ? C is a function bounded by 1 in magnitude and obeying ||f||U ? c, provided ? < p, one can find a polynomial ?: G ? Fp of degree at most k + ? ? 1 and functions gi : ?j?[k]\{i} Uj ? {z ? C: |z| ? 1} for i ? [k] such that |Ex?G f(x)??(x) ? i? [k] gi(x1,... ,xi?1, xi+1,..., xk)| ? (exp(Op,k,?(1))(Op,k,?(c?1)))?1. The proof relies on an approximation theorem for the cuboid-counting function that is proved using the inverse theorem for Freiman multi-homomorphisms.

A note on the generators of the polynomial algebra of six variables and application

"Let $\mathcal P_{n}:=H^{*}((\mathbb{R}P^{\infty})^{n}) \cong \mathbb Z_2[x_{1},x_{2},\ldots,x_{n}]$ be the graded polynomial algebra over $\mathcal K,$ where $\mathcal K$ denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra $\mathcal P_{n},$ viewed as a graded left module over the mod-$2$ Steenrod algebra, $\mathcal{A}.$ For $n>4,$ this problem is still unsolved, even in the case of $n=5$ with the help of computers. In this paper, we study the hit problem for the case $n=6$ in degree $d_{k}=6(2^{k} -1)+9.2^{k},$ with $k$ an arbitrary non-negative integer. By considering $\mathcal K$ as a trivial $\mathcal A$-module, then the hit problem is equivalent to the problem of finding a basis of $\mathcal K$-graded vector space $\mathcal K {\otimes}_{\mathcal{A}}\mathcal P_{n}.$ The main goal of the current paper is to explicitly determine an admissible monomial basis of the $\mathcal K$-graded vector space $\mathcal K{\otimes}_{\mathcal{A}}\mathcal P_6$ in some degrees. At the same time, the behavior of the sixth Singer algebraic transfer in degree $d_{k}=6(2^{k} -1)+9.2^{k}$ is also discussed at the end of this article. Here, the Singer algebraic transfer is a homomorphism from the homology of the mod-$2$ Steenrod algebra, $\mbox{Tor}^{\mathcal{A}}_{n, n+d}(\mathcal K, \mathcal K),$ to the subspace of $\mathcal K\otimes_{\mathcal{A}}\mathcal P_{n}$ consisting of all the $GL_n$-invariant classes of degree $d.$"

Decomposing Linear Layers

There are many recent results on reverse-engineering (potentially hidden) structure in cryptographic S-boxes. The problem of recovering structure in the other main building block of symmetric cryptographic primitives, namely, the linear layer, has not been paid that much attention so far. To fill this gap, in this work, we develop a systematic approach to decomposing structure in the linear layer of a substitutionpermutation network (SPN), covering the case in which the specification of the linear layer is obfuscated by applying secret linear transformations to the S-boxes. We first present algorithms to decide whether an ms x ms matrix with entries in a prime field Fp can be represented as an m x m matrix over the extension field Fps . We then study the case of recovering structure in MDS matrices by investigating whether a given MDS matrix follows a Cauchy construction. As an application, for the first time, we show that the 8 x 8 MDS matrix over F28 used in the hash function Streebog is a Cauchy matrix.

Leakage-Resilient Secret Sharing With Constant Share Size

In this work, we consider the leakage-resilience of algebraic-geometric (AG for short) codes based ramp secret sharing schemes extending the analysis on the leakage-resilience of linear threshold secret sharing schemes over prime fields that is done by Benhamouda et al. in the effort to construct linear leakage-resilient secret sharing schemes with constant share size. Since there does not exist any explicit efficient construction of AG codes over prime fields with constant field size, we consider constructions over prime fields with the help of concatenation method and constructions of codes over field extensions. Extending the Fourier analysis done by Benhamouda et al., one can show that concatenated algebraic geometric codes over prime fields do produce some nice leakage-resilient secret sharing schemes. One natural and curious question is whether AG codes over extension fields produce better leakage-resilient secret sharing schemes than the construction based on concatenated AG codes. Such construction provides several advantage compared to the construction over prime fields using concatenation method. It is clear that AG codes over extension fields give secret sharing schemes with a smaller reconstruction threshold for a fixed privacy parameter <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$t$ </tex-math></inline-formula> . In this work, it is also confirmed that indeed AG codes over extension fields have stronger leakage-resilience under some reasonable assumptions. Furthermore, we also show that AG codes over extension fields may provide strong multiplicative property which may be used in its application to the study of multiparty computation. In contrast, the same cannot be said for constructions based on concatenated AG codes, even when we are considering multiplication friendly embeddings. These advantages strongly motivate the study of secret sharing schemes from AG codes over extension fields. The current paper has two main contributions: (i) we obtain leakage-resilient secret sharing schemes with constant share sizes and unbounded numbers of players. Some of the schemes constructed without the use of concatenation also possesses strong multiplicative property (ii) via Fourier Analysis, we analyze the leakage-resilience of secret sharing schemes from codes over extension fields. This is of its own theoretical interest independent of its application to secret sharing schemes from algebraic geometric codes over extension fields.

Counting skew monomials in the Frobenius skew polynomial ring

We study the Frobenius skew polynomial ring over a polynomial ring in finitely many variables and show that growth function associated to the variables and the Frobenius satisfies a linear recurrence. We introduce the concept of Gelfand-Kirillov base for a finitely generated standard graded algebra over the prime field of characteristic p, where p is prime, and show its existence for the polynomial ring.

A Graphical Calculus for Lagrangian Relations

Symplectic vector spaces are the phase spaces of linear mechanical systems. The symplectic form describes, for example, the relation between position and momentum as well as current and voltage. The category of linear Lagrangian relations between symplectic vector spaces is a symmetric monoidal subcategory of relations which gives a semantics for the evolution -- and more generally linear constraints on the evolution -- of various physical systems. We give a new presentation of the category of Lagrangian relations over an arbitrary field as a `doubled' category of linear relations. More precisely, we show that it arises as a variation of Selinger's CPM construction applied to linear relations, where the covariant orthogonal complement functor plays the role of conjugation. Furthermore, for linear relations over prime fields, this corresponds exactly to the CPM construction for a suitable choice of dagger. We can furthermore extend this construction by a single affine shift operator to obtain a category of affine Lagrangian relations. Using this new presentation, we prove the equivalence of the prop of affine Lagrangian relations with the prop of qudit stabilizer theory in odd prime dimensions. We hence obtain a unified graphical language for several disparate process theories, including electrical circuits, Spekkens' toy theory, and odd-prime-dimensional stabilizer quantum circuits.

Lightweight ECC Coprocessor With Resistance Against Power Analysis Attacks Over NIST Prime Fields

This brief proposes a compact and fast elliptic curve cryptography coprocessor using affine coordinates. Although using affine coordinates requires compute-intensive modular inversion (ModInv), the proposed coprocessor can be accelerated using the improved ModInv. In addition, the study proposes a novel point multiplication method that is effective in preventing simple power analysis when affine coordinates are used. Using point randomization, the proposed coprocessor is also resistant to statistical power analysis (PA) such as differential PA (DPA) and correlation PA (CPA), and it requires 60 k gate counts and can perform one elliptic curve point multiplication within 1 ms at the maximum clock frequency.

On the Application of Noiseless Steganography and Elliptic Curves Cryptography Digital Signature Algorithm Methods in Securing Text Messages

Elliptic curve cryptography includes symmetric key cryptography systems that base their security on mathematical problems of elliptic curves. There are several ways that can be used to define the elliptic curve equation that depends on the infinite field used, one of which is the infinite field prima ( where ). Elliptic curve cryptography can be used for multiple protocol purposes, digital signatures, and encryption schemes.The purpose of this study is to determine the process of hiding encrypted messages using the Noiseless Steganography method as well as the generation of private keys and public keys and the process of verifying the validity of the Elliptic Curves Cryptography Digital Signature Algorithm (ECDSA). The result of this thesis is that a line graph is obtained that store or hides a message using the steganography method and a message authenticity from the process of key generation and verification of validity using the ECDSA method. By selecting three samples consisting of one test sample and two differentiating samples, a line graph, an MD5 hash value, and a value at the point are obtained different. Successively obtained values to message “Matematika 2018”, “MATEMATIKA 2018”, and “2018 matematika” are and . The discussion in this thesis only covers the elliptic curves on prime finite field. So, for the next thesis, the next researcher can do a discussion about the elliptic curve on the finite field or the application of elliptic curve cryptography and other steganographic methods

A note on the hit problem for the polynomial algebra of six variables and the sixth algebraic transfer

Let A be the classical, singly-graded Steenrod algebra over the prime order field F2 and let P⊗h:=F2[t1,…,th] denote the polynomial algebra on h generators, each of degree 1, viewed as a module over A. Write GLh for the usual general linear group of rank h over F2. As well known, the (mod 2) cohomology groups of the Steenrod algebra, ExtAh,h+⁎(F2,F2) is still largely mysterious for all homological degrees h≥6. The h-th algebraic transferTrhA:(F2⊗GLhAnnA‾[P⊗h]⁎)n⟶ExtAh,h+n(F2,F2), defined by William Singer [28], is a helpful tool to describe that Ext groups. Singer conjectured that this transfer is a monomorphism, but it remains open for any h≥5. There is currently no information on the conjecture for h=6. In this paper, we verify Singer's conjecture for homological degree h=6 and the internal degree of the general form ns:=6(2s−1)+10⋅2s. This result is important, since it tells us that the non-zero elements h22g1=h4Ph2∈ExtA6,6+n1(F2,F2), and D2∈ExtA6,6+n2(F2,F2) are not in the image of the sixth algebraic transfer. This Note is a continuation of our previous one [22], which will refer to as Part I. Our approach is based on explicitly solving the hit problem for the Steenrod algebra in the case h=6 and degree ns. This extends a result in Mothebe, Kaelo and Ramatebele [13]. At the same time, we also use this obtained result to establish the dimension result for the space of the unhit elements, F2⊗AP⊗7 in a certain general degree.

A point-conic incidence bound and applications over [formula omitted

In this paper, we prove the first incidence bound for points and conics over prime fields. As applications, we prove new results on expansion of bivariate polynomial images and on certain variations of distinct distances problems. These include new lower bounds on the number of pinned algebraic distances as well as improvements of results of Koh and Sun (2014) and Shparlinski (2006) on the size of the distance set formed by two large subsets of finite dimensional vector spaces over finite fields. We also prove a variant of Beck’s theorem for conics.

The Extremal Function for Excluding Geometry Minors over Prime Fields

A frame template over a field $\mathbb F$ describes the precise way in which a given $\mathbb F$-representable matroid is close to being a frame matroid. Our main result determines the maximum-rank projective or affine geometry that is described by a given frame template over a prime field. Subject to the matroid minors hypothesis of Geelen, Gerards, and Whittle, we use our result to determine, for each projective or affine geometry $N$ over a prime field $\mathbb F$, a best-possible upper bound on the number of elements in a simple $\mathbb F$-representable matroid $M$ of sufficiently large rank with no $N$-minor.

Equivalence between Varieties of Łukasiewicz–Moisil Algebras and Rings

Abstract The Post, axled and Łukasiewicz–Moisil algebras are important lattices studied in algebraic logic. In this paper, we investigate a useful interpretation between these algebras and some rings. We give a term equivalence between Post algebras of order $p$ and $p$-rings, $p$ prime and lift this result to the axled Łukasiewicz–Moisil algebra $L \cong B_s \times P$ and the ring $\prod ^s F_2 \times \prod ^l F_p$, where $B_s$ is a Boolean algebra of order $2^s$, $P$ a $p$-valued Post algebra of order $p^l$ and $F_p$ is the prime field of order $p$.

A refinement of Cauchy–Schwarz complexity

We introduce a notion of complexity for systems of linear forms called sequential Cauchy–Schwarz complexity, which is parametrized by two positive integers k,ℓ and refines the notion of Cauchy–Schwarz complexity introduced by Green and Tao. We prove that if a system of linear forms has sequential Cauchy–Schwarz complexity at most (k,ℓ) then any average of 1-bounded functions over this system is controlled by the 21−ℓ-th power of the Gowers Uk+1-norms of the functions. For ℓ=1 this agrees with Cauchy–Schwarz complexity, but for ℓ>1 there are systems that have sequential Cauchy–Schwarz complexity at most (k,ℓ) whereas their Cauchy–Schwarz complexity is greater than k. Our main application illustrates this with systems over a prime field Fp, denoted by Φk,M, which can be viewed as M-dimensional arithmetic progressions of length k. For each M≥2 we prove that Φk,M has sequential Cauchy–Schwarz complexity at most (k−2,|Φk,M|) (where |Φk,M| is the number of forms in the system), whereas the Cauchy–Schwarz complexity of Φk,M can be greater than k−2. Thus we obtain polynomial true-complexity bounds for Φk,M with exponent 2−|Φk,M|. A recent general theorem of Manners, proved independently with different methods, implies a similar application but with different polynomial true-complexity bounds, as explained in the paper. In separate work, we use our application to give a new proof of the inverse theorem for Gowers norms on Fpn, and related results on ergodic actions of Fpω.

Some fast algorithms multiplying a matrix by its adjoint

We present a non-commutative algorithm for the multiplication of a 2×2 block-matrix by its adjoint, defined by a matrix ring anti-homomorphism. This algorithm uses 5 block products (3 recursive calls and 2 general products). It works unconditionally in positive characteristic, or over the complex field with transposition, but for instance not over the complex field with conjugate-transpose. The resulting algorithm for arbitrary dimensions is a reduction of multiplication of a matrix by its adjoint to general matrix product, improving by a constant factor previously known reductions. We prove also that there is no algorithm derived from bilinear forms using only four products and the adjoint of one of them. Second we give novel dedicated algorithms for the complex field and the quaternions to compute the multiplication alternatively, taking advantage of the structure of the involved matrix-polynomial arithmetic. We then analyze the respective ranges of predominance of the two strategies. Finally we propose schedules with low memory footprint that support a fast and memory efficient practical implementation over a prime field.