Semilinear Transformations in Coding Theory: A New Technique in Code-Based Cryptography

This paper presents a new technique for disturbing the algebraic structure of linear codes in code-based cryptography. Specifically, we introduce the so-called semilinear transformations in coding theory and then apply them to the construction of code-based cryptosystems. Note that Fqm can be viewed as an Fq -linear space of dimension m , a semilinear transformation φ is therefore defined as an Fq -linear automorphism of Fqm . Then we impose this transformation to a linear code C over Fqm . It is clear that φ (C) forms an Fq -linear space, but generally does not preserve the Fqm -linearity any longer. Inspired by this observation, a new technique for masking the structure of linear codes is developed in this paper. Meanwhile, we endow the underlying Gabidulin code with the so-called partial cyclic structure to reduce the public-key size. Compared to some other code-based cryptosystems, our proposal admits a much more compact representation of public keys. For instance, 2592 bytes are enough to achieve the security of 256 bits, almost 403 times smaller than that of Classic McEliece entering the third round of the NIST PQC project.

Panoramic visual statistics shape retina-wide organization of receptive fields

Visual systems have adapted to the structure of natural stimuli. In the retina, center-surround receptive fields (RFs) of retinal ganglion cells (RGCs) appear to efficiently encode natural sensory signals. Conventionally, it has been assumed that natural scenes are isotropic and homogeneous; thus, the RF properties are expected to be uniform across the visual field. However, natural scene statistics such as luminance and contrast are not uniform and vary significantly across elevation. Here, by combining theory and novel experimental approaches, we demonstrate that this inhomogeneity is exploited by RGC RFs across the entire retina to increase the coding efficiency. We formulated three predictions derived from the efficient coding theory: (i) optimal RFs should strengthen their surround from the dimmer ground to the brighter sky, (ii) RFs should simultaneously decrease their center size and (iii) RFs centered at the horizon should have a marked surround asymmetry due to a stark contrast drop-off. To test these predictions, we developed a new method to image high-resolution RFs of thousands of RGCs in individual retinas. We found that the RF properties match theoretical predictions, and consistently change their shape from dorsal to the ventral retina, with a distinct shift in the RF surround at the horizon. These effects are observed across RGC subtypes, which were thought to represent visual space homogeneously, indicating that functional retinal streams share common adaptations to visual scenes. Our work shows that RFs of mouse RGCs exploit the non-uniform, panoramic structure of natural scenes at a previously unappreciated scale, to increase coding efficiency.

Expectation violations produce error signals in mouse V1

Repeated exposure to visual sequences changes the form of evoked activity in the primary visual cortex (V1). Predictive coding theory provides a potential explanation for this, namely that plasticity shapes cortical circuits to encode spatiotemporal predictions and that subsequent responses are modulated by the degree to which actual inputs match these expectations. Here we use a recently developed statistical modeling technique called Model-Based Targeted Dimensionality Reduction (MbTDR) to study visually-evoked dynamics in mouse V1 in context of a previously described experimental paradigm called "sequence learning". We report that evoked spiking activity changed significantly with training, in a manner generally consistent with the predictive coding framework. Neural responses to expected stimuli were suppressed in a late window (100-150ms) after stimulus onset following training, while responses to novel stimuli were not. Omitting predictable stimuli led to increased firing at the expected time of stimulus onset, but only in trained mice. Substituting a novel stimulus for a familiar one led to changes in firing that persisted for at least 300ms. In addition, we show that spiking data can be used to accurately decode time within the sequence. Our findings are consistent with the idea that plasticity in early visual circuits is involved in coding spatiotemporal information.

New classes of graphs with edge $ \; \delta- $ graceful labeling

<abstract><p>Graph labeling is a source of valuable mathematical models for an extensive range of applications in technologies (communication networks, cryptography, astronomy, data security, various coding theory problems). An edge $ \; \delta - $ graceful labeling of a graph $ G $ with $ p\; $ vertices and $ q\; $ edges, for any positive integer $ \; \delta $, is a bijective $ \; f\; $ from the set of edge $ \; E(G)\; $ to the set of positive integers $ \; \{ \delta, \; 2 \delta, \; 3 \delta, \; \cdots\; , \; q\delta\; \} $ such that all the vertex labels $ \; f^{\ast} [V(G)] $, given by: $ f^{\ast}(u) = (\sum\nolimits_{uv \in E(G)} f(uv)\; )\; mod\; (\delta \; k) $, where $ k = max (p, q) $, are pairwise distinct. In this paper, we show the existence of an edge $ \; \delta- $ graceful labeling, for any positive integer $ \; \delta $, for the following graphs: the splitting graphs of the cycle, fan, and crown, the shadow graphs of the path, cycle, and fan graph, the middle graphs and the total graphs of the path, cycle, and crown. Finally, we display the existence of an edge $ \; \delta- $ graceful labeling, for the twig and snail graphs.</p></abstract>

On classification of finite commutative chain rings

<abstract><p>Let $ R $ be a finite commutative chain ring with invariants $ p, n, r, k, m. $ It is known that $ R $ is an extension over a Galois ring $ GR(p^n, r) $ by an Eisenstein polynomial of some degree $ k $. If $ p\nmid k, $ the enumeration of such rings is known. However, when $ p\mid k $, relatively little is known about the classification of these rings. The main purpose of this article is to investigate the classification of all finite commutative chain rings with given invariants $ p, n, r, k, m $ up to isomorphism when $ p\mid k. $ Based on the notion of j-diagram initiated by Ayoub, the number of isomorphism classes of finite (complete) chain rings with $ (p-1)\nmid k $ is determined. In addition, we study the case $ (p-1)\mid k, $ and show that the classification is strongly dependent on Eisenstein polynomials not only on $ p, n, r, k, m. $ In this case, we classify finite (incomplete) chain rings under some conditions concerning the Eisenstein polynomials. These results yield immediate corollaries for p-adic fields, coding theory and geometry.</p></abstract>

Constructions and Necessities of Some Permutation Polynomials over Finite Fields

Let F q denote the finite field with q elements. Permutation polynomials over finite fields have important applications in many areas of science and engineering such as coding theory, cryptography, and combinatorial design. The study of permutation polynomials has a long history, and many results are obtained in recent years. In this paper, we obtain some further results about the permutation properties of permutation polynomials. Some new classes of permutation polynomials are constructed, and the necessities of some permutation polynomials are studied.

Post-Quantum Two-party Adaptor Signature Based on Coding Theory

An adaptor signature can be viewed as a signature concealed with a secret value and, by design, any two of the trio yield the other. In a multiparty setting, an initial adaptor signature allows each party create additional adaptor signatures without the original secret. Adaptor signatures help address scalability and interoperabity issues in blockchain. They can also bring some important advantages to cryptocurrencies, such as low on-chain cost, improved transaction fungibility, and less limitations of a blockchain&rsquo;s scripting language. In this paper, we propose a new two-party adaptor signature scheme that relies on quantum-safe hard problems in coding theory. The proposed scheme uses a hash-and-sign code-based signature scheme introduced by Debris-Alazard et al. and a code-based hard relation defined from the well-known syndrome decoding problem. To achieve all the basic properties of adaptor signatures formalized by Aumayr et al., we introduce further modifications to the aforementioned signature scheme. We also give a security analysis of our scheme and its application to the atomic swap. After providing a set of parameters for our scheme, we show that it has the smallest pre-signature size compared to existing post-quantum adaptor signatures.

An Efficient Coding Technique for Stochastic Processes

In the framework of coding theory, under the assumption of a Markov process (Xt) on a finite alphabet A, the compressed representation of the data will be composed of a description of the model used to code the data and the encoded data. Given the model, the Huffman’s algorithm is optimal for the number of bits needed to encode the data. On the other hand, modeling (Xt) through a Partition Markov Model (PMM) promotes a reduction in the number of transition probabilities needed to define the model. This paper shows how the use of Huffman code with a PMM reduces the number of bits needed in this process. We prove the estimation of a PMM allows for estimating the entropy of (Xt), providing an estimator of the minimum expected codeword length per symbol. We show the efficiency of the new methodology on a simulation study and, through a real problem of compression of DNA sequences of SARS-CoV-2, obtaining in the real data at least a reduction of 10.4%.