Algorithm 959: VBF: A Library of C plus plus Classes for Vector Boolean Functions in Cryptography

VBF is a collection of C++ classes designed for analyzing vector Boolean functions (functions that map a Boolean vector to another Boolean vector) from a cryptographic perspective. This implementation uses the NTL library from Victor Shoup, adding new modules that call NTL functions and complement the existing ones, making it better suited to cryptography. The class representing a vector Boolean function can be initialized by several alternative types of data structures such as Truth Table, Trace Representation, and Algebraic Normal Form (ANF), among others. The most relevant cryptographic criteria for both block and stream ciphers as well as for hash functions can be evaluated with VBF: it obtains the nonlinearity, linearity distance, algebraic degree, linear structures, and frequency distribution of the absolute values of the Walsh Spectrum or the Autocorrelation Spectrum, among others. In addition, operations such as equality testing, composition, inversion, sum, direct sum, bricklayering (parallel application of vector Boolean functions as employed in Rijndael cipher), and adding coordinate functions of two vector Boolean functions are presented. Finally, three real applications of the library are described: the first one analyzes the KASUMI block cipher, the second one analyzes the Mini-AES cipher, and the third one finds Boolean functions with very high nonlinearity, a key property for robustness against linear attacks

Isogenies of Elliptic Curves: A Computational Approach

Isogenies, the mappings of elliptic curves, have become a useful tool in cryptology. These mathematical objects have been proposed for use in computing pairings, constructing hash functions and random number generators, and analyzing the reducibility of the elliptic curve discrete logarithm problem. With such diverse uses, understanding these objects is important for anyone interested in the field of elliptic curve cryptography. This paper, targeted at an audience with a knowledge of the basic theory of elliptic curves, provides an introduction to the necessary theoretical background for understanding what isogenies are and their basic properties. This theoretical background is used to explain some of the basic computational tasks associated with isogenies. Herein, algorithms for computing isogenies are collected and presented with proofs of correctness and complexity analyses. As opposed to the complex analytic approach provided in most texts on the subject, the proofs in this paper are primarily algebraic in nature. This provides alternate explanations that some with a more concrete or computational bias may find more clear.Comment: Submitted as a Masters Thesis in the Mathematics department of the University of Washingto

Interpolation Cryptanalysis of Unbalanced Feistel Networks with Low Degree Round Functions

Arithmetisierungs-Orientierte Symmetrische Primitive (AOSPs) sprechen das bestehende Optimierungspotential bei der Auswertung von Blockchiffren und Hashfunktionen als Bestandteil von sicherer Mehrparteienberechnung, voll-homomorpher Verschlüsselung und Zero-Knowledge-Beweisen an. Die Konstruktionsweise von AOSPs unterscheidet sich von traditionellen Primitiven durch die Verwendung von algebraisch simplen Elementen. Zusätzlich sind viele Entwürfe über Primkörpern statt über Bits definiert. Aufgrund der Neuheit der Vorschläge sind eingehendes Verständnis und ausgiebige Analyse erforderlich um ihre Sicherheit zu etablieren. Algebraische Analysetechniken wie zum Beispiel Interpolationsangriffe sind die erfolgreichsten Angriffsvektoren gegen AOSPs. In dieser Arbeit generalisieren wir eine existierende Analyse, die einen Interpolationsangriff mit geringer Speicherkomplexität verwendet, um das Entwurfsmuster der neuen Chiffre GMiMC und ihrer zugehörigen Hashfunktion GMiMCHash zu untersuchen. Wir stellen eine neue Methode zur Berechnung des Schlüssels basierend auf Nullstellen eines Polynoms vor, demonstrieren Verbesserungen für die Komplexität des Angriffs durch Kombinierung mehrere Ausgaben, und wenden manche der entwickelten Techniken in einem algebraischen Korrigierender-Letzter-Block Angriff der Schwamm-Konstruktion an. Wir beantworten die offene Frage einer früheren Arbeit, ob die verwendete Art von Interpolationsangriffen generalisierbar ist, positiv. Wir nennen konkrete empfohlene untere Schranken für Parameter in den betrachteten Szenarien. Außerdem kommen wir zu dem Schluss dass GMiMC und GMiMCHash gegen die in dieser Arbeit betrachteten Interpolationsangriffe sicher sind. Weitere kryptanalytische Anstrengungen sind erforderlich um die Sicherheitsgarantien von AOSPs zu festigen

Attacks Against Filter Generators Exploiting Monomial Mappings

International audienceFilter generators are vulnerable to several attacks which have led to well-known design criteria on the Boolean filtering function. However , Rønjom and Cid have observed that a change of the primitive root defining the LFSR leads to several equivalent generators. They usually offer different security levels since they involve filtering functions of the form F (x k) where k is coprime to (2 n − 1) and n denotes the LFSR length. It is proved here that this monomial equivalence does not affect the resistance of the generator against algebraic attacks, while it usually impacts the resistance to correlation attacks. Most importantly, a more efficient attack can often be mounted by considering non-bijective mono-mial mappings. In this setting, a divide-and-conquer strategy applies based on a search within a multiplicative subgroup of F * 2 n. Moreover, if the LFSR length n is not a prime, a fast correlation involving a shorter LFSR can be performed

Fast algebraic immunity of Boolean functions and LCD codes

Nowadays, the resistance against algebraic attacks and fast algebraic attacks are considered as an important cryptographic property for Boolean functions used in stream ciphers. Both attacks are very powerful analysis concepts and can be applied to symmetric cryptographic algorithms used in stream ciphers. The notion of algebraic immunity has received wide attention since it is a powerful tool to measure the resistance of a Boolean function to standard algebraic attacks. Nevertheless, an algebraic tool to handle the resistance to fast algebraic attacks is not clearly identified in the literature. In the current paper, we propose a new parameter to measure the resistance of a Boolean function to fast algebraic attack. We also introduce the notion of fast immunity profile and show that it informs both on the resistance to standard and fast algebraic attacks. Further, we evaluate our parameter for two secondary constructions of Boolean functions. Moreover, A coding-theory approach to the characterization of perfect algebraic immune functions is presented. Via this characterization, infinite families of binary linear complementary dual codes (or LCD codes for short) are obtained from perfect algebraic immune functions. The binary LCD codes presented in this paper have applications in armoring implementations against so-called side-channel attacks (SCA) and fault non-invasive attacks, in addition to their applications in communication and data storage systems

Notes on Small Private Key Attacks on Common Prime RSA

We point out critical deficiencies in lattice-based cryptanalysis of common prime RSA presented in ``Remarks on the cryptanalysis of common prime RSA for IoT constrained low power devices'' [Information Sciences, 538 (2020) 54--68]. To rectify these flaws, we carefully scrutinize the relevant parameters involved in the analysis during solving a specific trivariate integer polynomial equation. Additionally, we offer a synthesized attack illustration of small private key attacks on common prime RSA.Comment: 15 pages, 1 figur

Cryptanalysis of a Theorem: Decomposing the Only Known Solution to the Big APN Problem

The existence of Almost Perfect Non-linear (APN) permutations operating on an even number of bits has been a long standing open question until Dillon et al., who work for the NSA, provided an example on 6 bits in 2009. In this paper, we apply methods intended to reverse-engineer S-Boxes with unknown structure to this permutation and find a simple decomposition relying on the cube function over GF(2^3) . More precisely, we show that it is a particular case of a permutation structure we introduce, the butterfly. Such butterflies are 2n-bit mappings with two CCZ-equivalent representations: one is a quadratic non-bijective function and one is a degree n+1 permutation. We show that these structures always have differential uniformity at most 4 when n is odd. A particular case of this structure is actually a 3-round Feistel Network with similar differential and linear properties. These functions also share an excellent non-linearity for n=3,5,7. Furthermore, we deduce a bitsliced implementation and significantly reduce the hardware cost of a 6-bit APN permutation using this decomposition, thus simplifying the use of such a permutation as building block for a cryptographic primitive

MinRoot: Candidate Sequential Function for Ethereum VDF

We present a candidate sequential function for a VDF protocol to be used within the Ethereum ecosystem. The new function, called MinRoot, is an optimized iterative algebraic transformation and is a strict improvement over competitors VeeDo and Sloth++. We analyze various attacks on sequentiality and suggest weakened versions for public scrutiny. We also announce bounties on certain research directions in cryptanalysis