Large Galois Research Articles

Enhancing image data security with chain and non-chain Galois ring structures

The local ring-based cryptosystem is built upon the core mathematical operations of the algebraic structure of local rings, which provides a significant advantage in ensuring security against advanced cryptanalysis. However, using the entire set of units within this structure would be computationally impractical in practical applications. Thus, we present a novel approach for designing 16×16 and 32×32 substitution boxes over chain and non-chain Galois ring respectively, by utilizing subgroups of local rings in a computationally feasible manner. Our proposed scheme significantly reduces memory usage by integrating both chain and non-chain rings. Specifically, the 16×16 and 32×32 S-boxes require only 16×28and 32×28 memory cells, respectively, whereas S-boxes of these dimensions over fields have been shown to be highly inefficient due to their extensive memory requirements (16×216and 32×232,respectively). The proposed method offers a more efficient solution for constructing S-boxes over large Galois fields and integrates chain and non-chain local Galois rings, as well as finite fields, for efficient transmission designs in smart devices. The algebraic structures used in this approach are used to establish the most vital aspect of a block cipher, the substitution box. We use each of these algebraic structures, each with a different bit size, to construct three distinct S-boxes. We discuss the performance and sensitivity of the proposed S-boxes to demonstrate their effectiveness in data protection. In this technique, the substitution boxes established by the non-chain ring, maximal cyclic subgroup of the Galois ring, and Galois field are applied for the processes of substitution, exclusive-or function, and diffusion in image encryption, respectively. Furthermore, we have performed numerous standard analyses on the encrypted image, including statistical analysis, differential analysis, and NIST tests. The results of these tests demonstrate the effectiveness of the proposed approach for various cryptographic purposes. Overall, this work provides a valuable contribution to the field of cryptography, particularly in constructing efficient S-boxes over large Galois fields.

Technology of protection of authentication data of computer network users

The article is devoted to the problem of perfect protection of authentication data of users of computer systems, especially in the case of a large number of different rights and powers that are personally granted to users. The importance of this problem increases especially when the number of such users in the system is hundreds or thousands. At the same time, each of them must create conditions for access only to their data and ensure protection against any outside interference, both by other users and by staff. A typical example of such conditions are secret electronic voting systems. The software and technical solutions described in this work have passed many years of testing and continue to be practically used in the electronic voting system of the Kyiv National University of Civil Engineering and Architecture. This system is regularly used for conducting elections to student self-government bodies and for surveys among students regarding the quality of teaching subjects. With the help of this system, elections of leaders of the Red Cross Society of Ukraine were held during the pandemic, which was associated with the Covid-19 virus. The mathematical basis of the described authentication data protection technology is the theory of algebraic groups, namely, the problem of discrete logarithmization over large Galois fields. Thanks to the use of cryptographic transformations over these fields, instead of known hash functions, it was possible to get rid of the possibility of collisions and neutralize the disclosure of passwords by attackers who were able to do it with the help of specialized Internet resources. It was also possible to increase resistance to brute-force password cracking, since conversions to Galois fields take ten times more time than hash function calculations.

Families of Abelian Varieties and Large Galois Images

Abstract Associated with an abelian variety $A$ of dimension $g$ over a number field $K$ is a Galois representation $\rho _A\colon \operatorname {Gal}({\overline {K}}/K)\to \operatorname {GL}_{2g}(\widehat {{\mathbb {Z}}})$. The representation $\rho _A$ encodes the Galois action on the torsion points of $A$ and its image is an interesting invariant of $A$ that contains a lot of arithmetic information. We consider abelian varieties over $K$ parametrized by the $K$-points of a non-empty open subvariety $U\subseteq {\mathbb {P}}^n_K$. We show that away from a set of density $0$, the image of $\rho _A$ will be very large; more precisely, it will have uniformly bounded index in a group obtained from the family of abelian varieties. This generalizes earlier results that assumed that the family of abelian varieties has “big monodromy”. We also give a version for families of abelian varieties with a more general base.

FSW: Fulcrum Sliding Window Coding for Low-Latency Communication

Fulcrum Random Linear Network Coding (RLNC) combines outer coding in a large Galois Field, e.g., GF(28), with inner coding in GF(2) to flexibly trade off the strong protection (low probability of linear dependent coding coefficients) of GF(28) with the low computational complexity of GF(2). However, the existing Fulcrum RLNC approaches are generation based, leading to large packet delays due to the joint processing of all packets in a generation in the encoder and decoder. In order to avoid these delays, we introduce Fulcrum Sliding Window (FSW) coding. We introduce two flavors of FSW: Fulcrum Nonsystematic Sliding Window (FNSW), which divides a given generation into multiple partially overlapping blocks, and Fulcrum Systematic Sliding Window (FSSW), which intersperses coded packets among the uncoded (systematic) transmission of the source packets in a generation. Our extensive evaluations indicate that FSSW substantially reduces the in-order packet delay (for moderately large generation and window sizes down to less than one fourth) and more than doubles the encoding and decoding (computation) throughput compared to generation-based Fulcrum.

Unlikely Intersections with ExCM curves in A2

The Zilber-Pink conjecture predicts that an algebraic curve in $\mathcal{A}_2$ has only finitely many intersections with the special curves, unless it is contained in a proper special subvariety. Under a large Galois orbits hypothesis, we prove the finiteness of the intersection with the special curves parametrising abelian surfaces isogenous to the product of two elliptic curves, at least one of which has complex multiplication. Furthermore, we show that this large Galois orbits hypothesis holds for curves satisfying a condition on their intersection with the boundary of the Baily-Borel compactification of $\mathcal{A}_2$. More generally, we show that a Hodge generic curve in an arbitrary Shimura variety has only finitely many intersection points with the generic points of a Hecke-facteur family, again under a large Galois orbits hypothesis.

The set of stable primes for polynomial sequences with large Galois group

Let K K be a number field with ring of integers O K \mathcal {O}_K , and let { f k } k ∈ N \{f_k\}_{k\in \mathbb {N}} be a sequence of monic polynomials in O K [ x ] \mathcal {O}_K[x] such that for every n ∈ N n\in \mathbb {N} , the composition f ( n ) = f 1 ∘ f 2 ∘ … ∘ f n f^{(n)}=f_1\circ f_2\circ \ldots \circ f_n is irreducible. In this paper we show that if the size of the Galois group of f ( n ) f^{(n)} is large enough (in a precise sense) as a function of n n , then the set of primes p ⊆ O K \mathfrak {p}\subseteq \mathcal {O}_K such that every f ( n ) f^{(n)} is irreducible modulo p \mathfrak {p} has density zero. Moreover, we prove that the subset of polynomial sequences such that the Galois group of f ( n ) f^{(n)} is large enough has density 1, in an appropriate sense, within the set of all polynomial sequences.

Large 2-adic Galois image and non-existence of certain abelian surfaces over ${\mathbb Q}$

Large 2-adic Galois image and non-existence of certain abelian surfaces over ${\mathbb Q}$

Arbitrarily large Galois orbits of non-homeomorphic surfaces

It has recently been proved that any element $$\sigma $$ of the absolute Galois group $$\mathrm{Gal}\,\overline{\mathbb {Q}}/{\mathbb Q}$$ different from the identity and complex conjugation conjugates some surface S into a surface $$S^{\sigma }$$ with non-isomorphic fundamental group. Here we use group theory to construct, for each integer $$n\geqslant 0$$ , an n-dimensional family of complex surfaces whose conjugates under $$\mathrm{Gal}\,\overline{\mathbb {Q}}/{\mathbb Q}$$ exhibit arbitrarily many non-isomorphic fundamental groups. These fundamental groups nevertheless have isomorphic profinite completions. The surfaces constructed are isogenous to higher products via actions of groups $$\mathrm{PGL}_2(p)$$ on algebraic curves.

Sufficient conditions for large Galois scaffolds

Let L/K be a finite, Galois, totally ramified p-extension of complete local fields with perfect residue fields of characteristic p>0. In this paper, we give conditions, valid for any Galois p-group G=Gal(L/K) (abelian or not) and for K of either possible characteristic (0 or p), that are sufficient for the existence of a Galois scaffold. The existence of a Galois scaffold makes it possible to address questions of integral Galois module structure, which is done in a separate paper [BCE]. But since our conditions can be difficult to check, we specialize to elementary abelian extensions and extend the main result of [Eld09] from characteristic p to characteristic 0. This result is then applied, using a result of Bondarko, to the construction of new Hopf orders over the valuation ring OK that lie in K[G] for G an elementary abelian p-group.

Large Galois images for Jacobian varieties of genus 3 curves

Given a prime number l greater than or equal to 5, we construct an infinite family of three-dimensional abelian varieties over Q such that, for any A/Q in the family, the Galois representation \rho_{A, l}: Gal_Q -> GSp(6, l) attached to the l-torsion of A is surjective. Any such variety A will be the Jacobian of a genus 3 curve over Q whose respective reductions at two auxiliary primes we prescribe to provide us with generators of Sp(6, l).

ON THE VANISHING OF COHOMOLOGIES OF <i>p</i>-ADIC GALOIS REPRESENTATIONS ASSOCIATED WITH ELLIPTIC CURVES

Let $K$ be a $p$-adic field and $E$ an elliptic curve over $K$ with potential good reduction. For some large Galois extensions $L$ of $K$ containing all $p$-power roots of unity, we show the vanishing of certain Galois cohomology groups of $L$ with values in the $p$-adic representation associated with $E$. This generalizes some results of Coates, Sujatha and Wintenberger.

Random Galois extensions of Hilbertian fields

Let L be a Galois extension of a countable Hilbertian field K. Although L need not be Hilbertian, we prove that an abundance of large Galois subextensions of L/K are.

Computing the Minimum Distance of Nonbinary LDPC Codes

Finding the minimum distance of low-density-parity-check (LDPC) codes is an NP-hard problem. Different from all existing works that focus on binary LDPC codes, we in this paper aim to compute the minimum distance of nonbinary LDPC codes, motivated by the fact that operating in a large Galois field provides one important degree of freedom to achieve both good waterfall and error-floor performance. Our method is based on the existing nearest nonzero codeword search (NNCS) method, but several modifications are incorporated for nonbinary LDPC codes, including the modified error impulse pattern, the dithering method, and the nonbinary decoder. Numerical results on the estimated minimum distances show that a code's minimum distance can be increased by careful selection of nonzero elements of the parity check matrix, or by increasing the mean column weight, or by increasing the size of the Galois field. These results support observations that have been made based on simulated performance in the literature. Finally, we provide an upper bound on the minimum distance for nonbinary quasi-cyclic LDPC codes.

On uniform large Galois images for modular abelian varieties

We formulate a question regarding uniform versions of "large Galois image properties" for modular abelian varieties of higher dimension, generalizing the well-known case of elliptic curves. We then answer our question affirmatively in the exceptional image case, and provide lower estimates for uniform bounds in the remaining cases.

Tamely Ramified Towers and Discriminant Bounds for Number Fields

The root discriminant of a number field of degree n is the nth root of the absolute value of its discriminant. Let R2m be the minimal root discriminant for totally complex number fields of degree 2m, and put α0 = lim infmR2m. One knows that α0 [ges ] 4πeγ ≈ 22.3, and, assuming the Generalized Riemann Hypothesis, α0 [ges ] 8πeγ ≈ 44.7. It is of great interest to know if the latter bound is sharp. In 1978, Martinet constructed an infinite unramified tower of totally complex number fields with small constant root discriminant, demonstrating that α0 < 92.4. For over twenty years, this estimate has not been improved. We introduce two new ideas for bounding asymptotically minimal root discriminants, namely, (1) we allow tame ramification in the tower, and (2) we allow the fields at the bottom of the tower to have large Galois closure. These new ideas allow us to obtain the better estimate α0 < 83.9.

Algebraic algorithms in GF(q)

This talk reports on joint work with R. Loos (Univ. Karlsruhe) on algebraic algorithms for computing in large Galois Fields GF(q) with q = p n where p is the characteristic of the field and may be arbitrarily large. This work is materialized by a module of algorithms implemented in the ALDES/SAC2 computer algebra system, which will be available with the next release of this system.

Kummer varieties and their Brauer groups

We study Kummer varieties attached to 2-coverings of abelian varieties of arbitrary dimension. Over a number field we show that the subgroup of odd order elements of the Brauer group does not obstruct the Hasse principle. Sufficient conditions for the triviality of the Brauer group are given, which allow us to give an example of a Kummer K3 surface of geometric Picard rank 17 over the rationals with trivial Brauer group. We establish the non-emptyness of the Brauer--Manin set of everywhere locally soluble Kummer varieties attached to 2-coverings of products of hyperelliptic Jacobians with large Galois action on 2-torsion.