The Journal of Slovak Academy of Sciences

Tatra Mountains Mathematical Publications

Tatra Mount. Mathe.Public.ONL

<abstract><title style='display:none'>Abstract</title>Let <italic>R</italic> be a principal ideal domain with quotient field <italic>K</italic>, and <italic>L</italic> = <italic>K</italic>(<italic>α</italic>), where <italic>α</italic> is a root of a monic irreducible polynomial <italic>F</italic> (<italic>x</italic>) ∈ <italic>R</italic>[<italic>x</italic>]. Let ℤ<italic>L</italic> be the integral closure of <italic>R</italic> in <italic>L</italic>. In this paper, for every prime <italic>p</italic> of <italic>R</italic>, we give a new efficient version of Dedekind’s criterion in <italic>R</italic>, i.e., necessary and sufficient conditions on <italic>F</italic> (<italic>x</italic>) to have <italic>p</italic> not dividing the index [ℤ<italic>L</italic>: <italic>R</italic>[<italic>α</italic>]], for every prime <italic>p</italic> of <italic>R</italic>. Some computational examples are given for <italic>R</italic> = ℤ.</abstract>

Abstract

Dept. of Math., Fac. of Sci. Dhar-El Mahraz

Keywords

MSC 2010

<article-title>Dedekind’s Criterion and Integral Bases</article-title>

Dedekind’s Criterion and Integral Bases

{"article-title":"Dedekind\u2019s Criterion and Integral Bases"}

<abstract><title style='display:none'>Abstract</title>In the first part we associate a periodic sequence with a partition and study the connection the distribution of elements of uniform limit of the sequences. Then some facts of statistical independence of these limits are proved.</abstract>

Department of Mathematics, Faculty of Education

<article-title>On Partitions and Periodic Sequences</article-title>

{"article-title":"On Partitions and Periodic Sequences"}

<abstract><title style='display:none'>Abstract</title>Extending Goursat’s Lemma we investigate the structure of subdirect products of 3-factor direct products. We construct several examples and then provide a structure theorem showing that every such group is essentially obtained by a combination of the examples. The central observation in this structure theorem is that the dependencies among the group elements in the subdirect product that involve all three factors are of Abelian nature. In the spirit of Goursat’s Lemma, for two special cases, we derive correspondence theorems between data obtained from the subgroup lattices of the three factors (as well as isomorphisms between arising factor groups) and the subdirect products. Using our results we derive an explicit formula to count the number of subdirect products of the direct product of three symmetric groups.</abstract>

<article-title>Subgroups of 3-Factor Direct Products</article-title>

{"article-title":"Subgroups of 3-Factor Direct Products"}

<abstract><title style='display:none'>Abstract</title>The most commonly used methods for solving classical (historical) ciphers are based on global optimization (meta-heuristic methods). Despite the fact that global optimization is a well-studied problem, in the case of classical ciphers, there are still many open questions such as the construction of fitness functions or efficient transformation of the cryptanalysis (breaking attempt) to an optimization problem. Therefore the transformation of a cryptanalytical task to an optimization problem and the choice of a suitable fitness function form an important part of the topic. In this paper, we focus on the simple columnar transposition in depth. Our main contribution is a detailed analysis and comparison of different fitness functions, fitness landscape analysis and solving experiments.</abstract>

Institute of Computer Science and Mathematics

Slovak University of Technology in Bratislava

<article-title>Cryptanalysis of the Columnar Transposition Using Meta-Heuristics</article-title>

Cryptanalysis of the Columnar Transposition Using Meta-Heuristics

{"article-title":"Cryptanalysis of the Columnar Transposition Using Meta-Heuristics"}

<abstract><title style='display:none'>Abstract</title>T-310 is an important Cold War cipher. The cipher is extremely complex and it outputs extremely few bits from the internal state. A recent paper [C<sc>ourtois</sc>, N. T.: <italic>Decryption oracle slide attacks on T-310</italic>, Cryptologia, <bold>42</bold> (2018), no. 3, 191–204] shows an example of a highly anomalous key such that T-310 can be broken by a slide attack with a decryption oracle. In this paper, we show that the same attacks are ALSO possible for regular keys which satisfy all the official KT1 requirements. Two other recent papers [C<sc>ourtois</sc>, N. T.—G<sc>eorgiou</sc>, M.—S<sc>carlata</sc>, M.: <italic>Slide attacks and LC-weak keys in T-310</italic>, Cryptologia <bold>43</bold> (2019), no. 3, 175–189]; [C<sc>ourtois</sc>, N. T.—O<sc>prisanu</sc>, M. B.—S<sc>chmeh</sc>, K.: <italic>Linear cryptanalysis and block cipher design in East Germany in the 1970s</italic>, Cryptologia (published online), December 5, 2018] show that some of the KT1 keys are very weak w.r.t. Linear Cryptanalysis. In this paper, we show that a vast number of such weak keys exist and study the exact pre-conditions which make them weak. In addition we introduce a new third class of weak keys for RKDC (Related-Key Differential Cryptanalysis). We show that the original designers in the 1970s have ensured that these RKDC properties cannot happen for 4 rounds. We have discovered that these properties can happen for as few as 5 rounds for some keys, and for 10 to 16 rounds they become hard to avoid. The main reason why we study weak keys is to show that none of these properties occur by accident, rather that they are governed by precise pre-conditions which guarantee their existence, and countless other keys with the same properties exist. Eventually, this is how interesting attacks can be found.</abstract>

ETH Zurich, Department of Computer Science

<article-title>How Many Weak-Keys Exist in T-310 ?</article-title>

{"article-title":"How Many Weak-Keys Exist in T-310 ?"}

<abstract><title style='display:none'>Abstract</title>In this paper we introduce a new cryptographic system which is based on the idea of encryption due to [McEliece, R. J. <italic>A public-key cryptosystem based on algebraic coding theory</italic>, DSN Progress Report. <bold>44</bold>, 1978, 114–116]. We use the McEliece encryption system with a new linear error-correcting code, which was constructed in [Hannusch, C.—Lakatos, P.: <italic>Construction of self-dual binary</italic> 22<italic>k,</italic> 22<italic>k</italic>−1, 2<italic>k-codes</italic>, Algebra and Discrete Math. <bold>21</bold> (2016), no. 1, 59–68]. We show how encryption and decryption work within this cryptosystem and we give the parameters for key generation. Further, we explain why this cryptosystem is a promising post-quantum candidate.</abstract>

<article-title>A Cryptographic System Based on a New Class of Binary Error-Correcting Codes</article-title>

A Cryptographic System Based on a New Class of Binary Error-Correcting Codes

{"article-title":"A Cryptographic System Based on a New Class of Binary Error-Correcting Codes"}

<abstract><title style='display:none'>Abstract</title>We investigate the information ratio of graph-based secret sharing schemes. This ratio characterizes the efficiency of a scheme measured by the amount of information the participants must remember for each bits in the secret.We examine the information ratio of several systems based on graphs with many leaves, by proving non-trivial lower and upper bounds for the ratio. On one hand, we apply the so-called entropy method for proving that the lower bound for the information ratio of <italic>n</italic>-sunlet graphs composed of a 1-factor between the vertices of a cycle <italic>Cn</italic> and <italic>n</italic> independent vertices is 2. On the other hand, some symmetric and recursive constructions are given that yield the upper bounds. In particular, we show that the information ratio of every graph composed of a 1-factor between a complete graph <italic>Kn</italic> and at most 4 independent vertices is smaller than 2.</abstract>

3in Research Group, Faculty of Informatics

<article-title>On the Information Ratio of Graphs with Many Leaves</article-title>

On the Information Ratio of Graphs with Many Leaves

{"article-title":"On the Information Ratio of Graphs with Many Leaves"}

<abstract><title style='display:none'>Abstract</title>In this paper we study two-round key-alternating block ciphers with round function <italic>f</italic> (<italic>x</italic>)= <italic>x</italic>(2<italic>t</italic>+1)2<italic>s</italic>, where <italic>t, s</italic> are positive integers. An algorithm to compute the distribution weight in respect to input and output masks is described. Also, in the case <italic>t</italic> = 1 the correlation distributions depending on input and output masks are completely determined for arbitrary pairs of masks.</abstract>

<article-title>Correlation Distribution Analysis of a Two-Round Key-Alternating Block Cipher</article-title>

Correlation Distribution Analysis of a Two-Round Key-Alternating Block Cipher

{"article-title":"Correlation Distribution Analysis of a Two-Round Key-Alternating Block Cipher"}

<abstract><title style='display:none'>Abstract</title>The recent Meltdown and Spectre vulnerabilities have highlighted a very present and real threat in the on-chip memory cache units which can ultimately provide a hidden state, albeit only readable via memory timing instructions [Kocher, P.—Genkin, D.— Gruss, D.— Haas, W.—Hamburg, M.—Lipp, M.–Mangard, S.—Prescher, T.—Schwarz, M.—Yarom, Y.: <italic>Spectre attacks: Exploiting speculative execution,</italic> CoRR, abs/1801.01203, 2018]. Yet the exploits, although having some complexity and slowness, are demonstrably reliable on nearly all processors produced for the last two decades.Moving out from looking at this strictly as a means of reading protected memory, as the large microprocessor companies move to close this security vulnerability, an interesting question arises. Could the inherent design of the processor give the ability to hide arbitrary calculations in this speculative and parallel side channel? Without even using protected memory and exploiting the vulnerability, as has been the focus, there could very well be a whole class of techniques which exploit the side-channel. It could be done in a way which would be largely un-preventable behavior as the technology would start to become self-defeating or require a more complicated and expensive on-chip cache memory system to properly post-speculatively clean itself. And the ability to train the branch predictor to incorrectly speculatively behave is almost certain given hardware limitations, andthusprovidesexactly this pathway.A novel approach looks at just how much computation can be done speculatively with a result store via indirect reads and available through the memory cache. A multi-threaded approach can allow a multi-stage computation pipeline where each computation is passed to a read-out thread and then to the next computation thread [Swanson, S.—McDowell, L. K.—Swift, M. M.—Eggers, S. J.–Levy H. M.: <italic>An evaluation of speculative instruction execution on simultaneous multithreaded processors,</italic> ACM Trans. Comput. Syst. <bold>21</bold> (2003), 314–340].Through channels like this, an application can surreptitiously make arbitrary calculations, or even leak data without any standard tracing tools being capable of monitoring the subtle changes. Like a variation of the famous physics Heisenberg uncertainty principle, even a tool capable of reading the cache states would not only be incredibly inefficient, but thereby tamper with and modify the state. Tools like in-circuit emulators, or specially designed cache emulators would be needed to unmask the speculative reads, and it is further difficult to visualize with a linear time-line.Specifically, the AES and RSA algorithms will be studied with respect to these ideas, looking at success rates for various calculation batches with speculative execution, while having a summary view to see the rather severe performance penalties for using such methods.Either approaches could provide for strong white-box cryptography when considering a binary, non-source code form. In terms of white-box methods, both could be significantly challenging to locate or deduce the inner workings of the code. Further, both methods can easily surreptitiously leak or hide data within shared memory in a seemingly innocuous manner.</abstract>

Eötvös Loránd Tudomány Egyetem University

<article-title>Self-Spectre, Write-Execute and the Hidden State</article-title>

Self-Spectre, Write-Execute and the Hidden State

{"article-title":"Self-Spectre, Write-Execute and the Hidden State"}

<abstract><title style='display:none'>Abstract</title>In the era of global informatization, transmitting and storing information in digital form it is very important to ensure an adequate level of security of ciphers used. Cryptanalysis deals with studying the level of security, thus exposing the weakness of theoretical and implemented cryptographic solutions. In this paper cryptanalysis of stream cipher VMPC using Tabu Search is shown. From estimates made on a full version of VMPC cipher we concluded that about 2157 possibilities needs to be checked in order to find the proper one, which would be the best attack known so far.</abstract>

<article-title>Tabu Cryptanalysis of VMPC Stream Cipher</article-title>

{"article-title":"Tabu Cryptanalysis of VMPC Stream Cipher"}

<abstract><title style='display:none'>Abstract</title>Fully homomorphic encryption (FHE) has been among the most popular research topics of the last decade. While the bootstrapping-based, public key cryptosystems that follow Gentry’s original design are getting more and more efficient, their performance is still far from being practical. This leads to several attempts to construct <italic>symmetric</italic> FHE schemes that would not be as inefficient as their public key counterparts. Unfortunately, most such schemes were also based on (randomized) linear transformations, and shown to be completely insecure. One such broken scheme was the Matrix Operation for Randomization and Encryption (MORE). In a recent paper, Hariss, Noura and Samhat propose Enhanced MORE, which is supposed to improve over MORE’s weaknesses. We analyze Enhanced MORE, discuss why it does not improve over MORE, and show that it is even less secure by presenting a highly efficient ciphertext-only decryption attack. We implement the attack and confirm its correctness.</abstract>

<article-title>Cryptanalysis of Enhanced More</article-title>

{"article-title":"Cryptanalysis of Enhanced More"}

<abstract><title style='display:none'>Abstract</title>BIKE suite of algorithms is one of the candidates in NIST call for public-key post-quantum cryptographic algorithms. It is a key-encapsulation mechanism based on QC-MDPC codes with purely ephemeral keys. The security device implementing such an algorithm therefore needs to generate multiple key pairs in its lifetime very efficiently.In our paper we explore the situation where BigBrother-type adversary can subtly corrupt the vendor(s) of the security devices (e.g., by altering the standard algorithms). In our model, BigBrother cannot preload the keys or synchronize the key generator by a covert channel, but is able to learn the secrets of security devices by observing the public execution of the KEM protocols. BigBrother typically obtains the secret through the usage of (masked) weak keys. However, we can also imagine other covert channels embedded into the ephemeral public keys by some unknown algorithm.To prevent these classes of attacks, we propose that the standard should explicitly specify a verifiable algorithm to transform the required key randomness into a set of keys.</abstract>

<article-title>Preventing Potential Backdoors in Bike Algorithm</article-title>

Preventing Potential Backdoors in Bike Algorithm

{"article-title":"Preventing Potential Backdoors in Bike Algorithm"}

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Honorary Editors † Beloslav Riečan † Miloslav Duchoň Editor in Chief Karol Nemoga Managing Editors Tibor Žáčik Branislav Novotný Editorial Board Jiří Anděl, Praha Ján Borsík, Košice Ferdinand Chovanec, Liptovský Mikuláš Anatolij Dvurečenskij, Bratislava Roman Frič, Košice Martin Gavalec, Hradec Králové Viliam Geffert, Košice Otokar Grošek, Bratislava Petr Hájek, Praha Ján Haluška, Košice Ľubica Holá, Bratislava Vladimír Janiš, Banská Bystrica Stanislav Jendroľ, Košice Kamil John, Praha Jozef Kačúr, Bratislava Martin Kalina, Bratislava Erich P. Klement, Linz Anna Kolesárová, Bratislava Helmut Länger, Vienna Peter Maličký, Banská Bystrica Radko Mesiar, Bratislava Mirko Navara, Praha Roman Nedela, Banská Bystrica Vilém Novák, Ostrava-Poruba Stanislav Palúch, Žilina Endré Pap, Novi Sad Andrej Pázman, Bratislava Miroslav Ploščica, Košice Pavel Pták, Praha Sylvia Pulmannová, Bratislava Zdenka Riečanová , Bratislava Peter Vojtáš, Košice Gejza Wimmer, Banská Bystrica Publisher De Gruyter Poland Bogumiła Zuga 32A Str. 01-811 Warsaw, Poland

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