On the Design of LIL Tests for (Pseudo) Random Generators and Some Experimental Results

NIST SP800-22 (2010) proposes the state of art testing suite for (pseudo) random generators to detect deviations of a binary sequence from randomness. On the one hand, as a counter example to NIST SP800-22 test suite, it is easy to construct functions that are considered as GOOD pseudorandom generators by NIST SP800-22 test suite though the output of these functions are easily distinguishable from the uniform distribution. Thus these functions are not pseudorandom generators by definition. On the other hand, NIST SP800-22 does not cover some of the important laws for randomness. Two fundamental limit theorems about random binary strings are the central limit theorem and the law of the iterated logarithm (LIL). Several frequency related tests in NIST SP800-22 cover the central limit theorem while no NIST SP800-22 test covers LIL. This paper proposes techniques to address the above challenges that NIST SP800-22 testing suite faces. Firstly, we propose statistical distance based testing techniques for (pseudo) random generators to reduce the above mentioned Type II errors in NIST SP800-22 test suite. Secondly, we propose LIL based statistical testing techniques, calculate the probabilities, and carry out experimental tests on widely used pseudorandom generators by generating around 30TB of pseudorandom sequences. The experimental results show that for a sample size of 1000 sequences (2TB), the statistical distance between the generated sequences and the uniform distribution is around 0.07 (with $0$ for statistically indistinguishable and $1$ for completely distinguishable) and the root-mean-square deviation is around 0.005

Best Effort and Practice Activation Codes

Activation Codes are used in many different digital services and known by many different names including voucher, e-coupon and discount code. In this paper we focus on a specific class of ACs that are short, human-readable, fixed-length and represent value. Even though this class of codes is extensively used there are no general guidelines for the design of Activation Code schemes. We discuss different methods that are used in practice and propose BEPAC, a new Activation Code scheme that provides both authenticity and confidentiality. The small message space of activation codes introduces some problems that are illustrated by an adaptive chosen-plaintext attack (CPA-2) on a general 3-round Feis- tel network of size 2^(2n) . This attack recovers the complete permutation from at most 2^(n+2) plaintext-ciphertext pairs. For this reason, BEPAC is designed in such a way that authenticity and confidentiality are in- dependent properties, i.e. loss of confidentiality does not imply loss of authenticity.Comment: 15 pages, 3 figures, TrustBus 201

Almost Optimal Pseudorandom Generators for Spherical Caps

Halfspaces or linear threshold functions are widely studied in complexity theory, learning theory and algorithm design. In this work we study the natural problem of constructing pseudorandom generators (PRGs) for halfspaces over the sphere, aka spherical caps, which besides being interesting and basic geometric objects, also arise frequently in the analysis of various randomized algorithms (e.g., randomized rounding). We give an explicit PRG which fools spherical caps within error $\epsilon$ and has an almost optimal seed-length of $O(\log n + \log(1/\epsilon) \cdot \log\log(1/\epsilon))$. For an inverse-polynomially growing error $\epsilon$, our generator has a seed-length optimal up to a factor of $O( \log \log {(n)})$. The most efficient PRG previously known (due to Kane, 2012) requires a seed-length of $\Omega(\log^{3/2}{(n)})$ in this setting. We also obtain similar constructions to fool halfspaces with respect to the Gaussian distribution. Our construction and analysis are significantly different from previous works on PRGs for halfspaces and build on the iterative dimension reduction ideas of Kane et. al. (2011) and Celis et. al. (2013), the \emph{classical moment problem} from probability theory and explicit constructions of \emph{orthogonal designs} based on the seminal work of Bourgain and Gamburd (2011) on expansion in Lie groups.Comment: 28 Pages (including the title page

Covers and Logarithmic Signatures of Finite Groups in Cryptography

After the first time Diffie and Hellmann [1] introduced the idea of separate keys, asymmetric cryptography has became increasingly developing. Many public key cryptosystems have been proposed, but only few of such systems remain unbroken. The most of them used nowadays are based on the perceived intractability of certain mathematical problems in very large, finite cyclic groups. In the late 1970's S. Magliveras started to investigate the use of special factorizations, called logarithmic signatures, of finite non-abelian groups in cryptography [2,3,4,5]. Later, Magliveras, Stinson and Tran van Trung [6] have done some preliminary work in creating two public key cryptosystems, MST1, based on logarithmic signatures, and MST2, based on another type of group coverings called [s,r]-meshes. Until now however, no practical realizations are known for MST1 or MST2. Recently, a new type of public key cryptosystem, called MST3 [7], has been developed on the basis of logarithmic signatures and random covers of finite non-abelian groups (i.e. factorization sequences in which blocks are constructed by sampling uniformly at random on the underlying group). For a possible realization of the generic version of this system, the Suzuki 2-groups have been suggested. %Due to their simple structure, these groups make it possible for studying the security of the scheme. The primary objective of this thesis is to show that cryptosystem MST3 can be realized with Suzuki 2-groups. To this question we can give an affirmative answer. There are several challenges in designing the practical realization of the scheme. The first problem is to efficiently generate covers for large groups and with good cryptographic properties. Showing the connection of this problem with the classical occupancy problem, we determine a bound for the probability that randomly chosen collection of group elements compose a cover. As a consequence, we solve the problem of generating random covers for arbitrary large groups. We also present several experimental computer results about covers and uniform covers for some alternating groups.Due to their simple structure, the Suzuki 2-groups enable us to study the security of the system and also provide an efficient implementation. In the first realization, a special class of canonical logarithmic signatures for elementary abelian 2-groups has been proposed as a basis for the key generation. These are easily constructed and allow highly efficient factorization. We provide an attack, showing that canonical signatures cannot be used to build secure realization of MST3 with Suzuki 2-groups. Motivated by the attack on the first realization, we propose a new variant with significant improvement, strengthening the system's security. For that purpose we re-design the set-up of the scheme and introduce a new class of fused transversal logarithmic signatures. These allow efficient factorization if we keep track of the transformations used to generate them. We present a thorough study of the security of the scheme by using heuristic and algebraic methods. We first determine the complexity for the lower bounds of conceivable direct attacks to recover the private key in terms of the size of the groups. These bounds give a hint of the strength of the system. We further develop a powerful method for a chosen plaintext attack showing non-fused transversal logarithmic signatures cannot be used. Moreover, proposed class of fused transversal logarithmic signatures withstand this attack when used in MST3 with Suzuki 2-groups and thus to our knowledge could be used to build secure realization of the scheme. We describe and discuss the implementation issues of the system in detail and include data of its performance obtained from an experimental result. Apart from the main research objective, we introduce a new approach to designing pseudorandom number generators based on random covers of finite groups. PRNGs based on random covers, called MSTg, turn out to be highly efficient for a certain class of group and produces high-quality random bit sequences. A very extensive sequence of tests for randomness using the NIST Statistical Test Suite and Diehard Battery of Tests provided here show extremely strong properties for the new methodology. More importantly, we show evidence that this class of generators is suitable for cryptographic applications. Finally, we include performance data of the generators and propose a method of using them in practice. [1] W. Diffie and M. E. Hellman, New Directions in Cryptography, IEEE Trans. on Inform. Theory, IT-22(6) (1976), 644–654. [2] S. S. Magliveras, B. A. Oberg and A. J. Surkan, A New Random Number Generator from Permutation Groups, In Rend. del Sem. Matemat. e Fis. di Milano, LIV (1984), 203–223. [3] S. S. Magliveras, A cryptosystem from logarithmic signatures of finite groups, in Proceedings of the 29’th Midwest Symposium on Circuits and Systems, Elsevier Publ. Co. (1986), 972–975. [4] S. S. Magliveras and N.D. Memon, Properties of Cryptosystem PGM, Advances in Cryptology, Lecture Notes in Comp. Sc., Springer-Verlag, 435 (1989), 447–460. [5] S. S. Magliveras and N.D. Memon, Random Permutations from Logarithmic Signatures, Computing in the 90’s, First Great Lakes Comp. Sc. Conf., Lecture Notes in Computer Science, Springer-Verlag, 507 (1989), 91–97. [6] S. S. Magliveras, Tran van Trung and D.R. Stinson, New approaches to designing public key cryptosystems using one-way functions and trap-doors in finite groups, J. of Cryptology, 15 (2002), 285–297. [7] W. Lempken, S. S. Magliveras, Tran van Trung and W. Wei, A public key cryptosystem based on non-abelian finite groups, J. of Cryptology, 22 (2009), 62–74.Nachdem Diffie und Hellman [1] die Idee von getrennten Schlüsseln für Verschlüsselungsverfahren präsentierten, wurde die asymmetrische Kryptographie zunehmend weiter entwickelt. Viele Public Key Kryptosysteme wurden vorgeschlagen, aber nur wenige wurden letztlich nicht gebrochen. Die meisten von ihnen, die noch heute verwendet werden, basieren auf den bekannten Schwierigkeiten von bestimmten mathematischen Problemen in sehr großen endlichen zyklischen Gruppen. In den späten 1970ern begann S. Magliveras den Nutzen spezieller Faktorisierungen auf endlichen nicht-abelschen Gruppen, bekannt als logarithmische Signaturen, in der Kryptographie zu erforschen [2,3,4,5]. Später folgten weitere wegweisende Arbeiten von Magliveras, Stinson und Tran van Trung [6] die sowohl das Kryptosystem MST1, welches auf logarithmischen Signaturen basiert, als auch MST2, das auf einer anderen Art von Gruppen-Überdeckungen – den sogenannten [s,r]-Gittern – arbeitet, bekannt machten. Bisher sind allerdings noch keine praktische Realisierungen von MST1 oder MST2 bekannt. Kürzlich wurde ein neues Public Key Kryptosystem namens MST3 [7] entwickelt, das auf der Grundlage von logarithmischen Signaturen und zufälligen Überdeckungen von endlichen nicht-abelschen Gruppen arbeitet. Für eine mögliche Realisierung der generischen Version dieses Systems wurden die Suzuki-2-Gruppen vorgeschlagen. Das Hauptziel dieser Arbeit liegt darin zu zeigen, dass MST3 auf Suzuki-2-Gruppen realisiert werden kann. Diese Frage können wir im positiven Sinne beantworten. Es gab einige Änderungen in der Umsetzung der Realisierung des Systems. Das erste Problem besteht darin, effizient zufällige Überdeckungen für große Gruppen mit guten kryptographischen Eigenschaften zu erzeugen. In dem wir den Bezug zum klassischen Belegungsproblem (“the occupancy problem”) herstellen, können wir eine Schranke für die Wahrscheinlichkeit, dass eine zufällige Ansammlung von Gruppenelementen eine Überdeckung bilden, bestimmen. Eine Konsequenz daraus ist, dass wir das Problem, zufällige Überdeckungen für beliebige große Gruppen zu erzeugen, lösen können. Weiterhin stellen wir einige Resultate spezieller Computerexperimente bezüglich Überdeckungen und gleichmäßigen Überdeckungen zu verschiedenen Gruppen vor. Dank ihrer einfachen Struktur erlauben uns die Suzuki-2-Gruppen die Sicherheit des Systems genau zu studieren und es effizient zu implementieren. In der ersten Realisierung wird eine spezielle Klasse von kanonisch logarithmischen Signaturen zu elementar-abelschen 2-Gruppen als Basis für die Schlüsselgenerierung verwendet. Diese sind leicht zu konstruieren und erlauben eine sehr effiziente Faktorisierung. Wir betrachten einen Angriff, der zeigt, dass kanonische Signaturen nicht benutzt werden können um eine sichere Umsetzung von MST3 mit Suzuki-2-Gruppen zu realisieren. Motiviert durch die Attacke auf die erste Realisierung konnten wir eine neue Variante mit signifikanten Verbesserungen vorstellen, welche die Sicherheit des Systems deutlich stärken. Zu diesem Zweck verwendeten wir für das Setup des Systems eine Funktion zur Maskierung des privaten Schlüssels. Ferner führten wir eine Klasse von fusionierten transversalen logarithmischen Signaturen für die Realisierung des Verfahrens ein. Diese erlauben eine effiziente Faktorisierung mit Hilfe einer “Trapdoor” Information. Wir stellen eine genaue Studie der Sicherheit des Systems vor, in dem wir heuristische und algebraische Methoden verwenden. Zunächst bestimmen wir die untere Schranke der Komplexität bezüglich der Gruppengröße von möglich vorstellbaren direkten Attacken, um den privaten Schlüssel zu erhalten. Diese Schranken geben einen Hinweis auf die Stärke des Systems. Weiterhin entwickeln wir eine mächtige Methode für eine Chosen-Plaintext-Attacke, und zeigen, dass nicht-fusionierte transversale logarithmische Signaturen nicht verwendet werden können. Zudem zeigen wir, dass die vorgeschlagene Klassen von fusionierten transversalen Signaturen dieser Attacke widerstehen, und nach unserem Wissen, sie damit eine sichere Realisierung des Systems ermöglichen. Wir beschreiben und diskutieren die Implementierung des Systems im Detail und ziehen dabei Daten über die Effizienz, die wir als Resultate von einem Experiment erhielten, mit ein. Abgesehen von dem zentralen Forschungsobjekt werden wir noch einen neuen Ansatz für die Konstruktion pseudo-zufälliger Zahlengeneratoren (PRNG) vorstellen, welcher auf zufälligen Überdeckungen von endlichen Gruppen basiert. PRNGs basierend auf zufälligen Überdeckungen, auch MSTg genannt, zeigten sich bisher zu einer bestimmten Klasse von Gruppen als höchst effizient und produzierten qualitativ hochwertige zufällige Bit-Sequenzen. Eine sehr komplexe Folge von aufwendigen Zufälligkeits-Tests zeigte durch Nutzung der NIST Statistical Test Suite und Diehard Battery of Test die starken Eigenschaften der neuen Methodik. Noch wichtiger ist allerdings, dass wir Beweise erbringen können, dass diese Klasse von Generatoren adäquat für kryptographische Anwendungen sind. Schließlich fügen wir noch Daten über die Effizienz der Generatoren an und schlagen eine Methode zur praktischen Anwendung vor. [1] W. Diffie and M. E. Hellman, New Directions in Cryptography, IEEE Trans. on Inform. Theory, IT-22(6) (1976), 644–654. [2] S. S. Magliveras, B. A. Oberg and A. J. Surkan, A New Random Number Generator from Permutation Groups, In Rend. del Sem. Matemat. e Fis. di Milano, LIV (1984), 203–223. [3] S. S. Magliveras, A cryptosystem from logarithmic signatures of finite groups, in Proceedings of the 29’th Midwest Symposium on Circuits and Systems, Elsevier Publ. Co. (1986), 972–975. [4] S. S. Magliveras and N.D. Memon, Properties of Cryptosystem PGM, Advances in Cryptology, Lecture Notes in Comp. Sc., Springer-Verlag, 435 (1989), 447–460. [5] S. S. Magliveras and N.D. Memon, Random Permutations from Logarithmic Signatures, Computing in the 90’s, First Great Lakes Comp. Sc. Conf., Lecture Notes in Computer Science, Springer-Verlag, 507 (1989), 91–97. [6] S. S. Magliveras, Tran van Trung and D.R. Stinson, New approaches to designing public key cryptosystems using one-way functions and trap-doors in finite groups, J. of Cryptology, 15 (2002), 285–297. [7] W. Lempken, S. S. Magliveras, Tran van Trung and W. Wei, A public key cryptosystem based on non-abelian finite groups, J. of Cryptology, 22 (2009), 62–74

Correlated Pseudorandomness from the Hardness of Quasi-Abelian Decoding

Secure computation often benefits from the use of correlated randomness to achieve fast, non-cryptographic online protocols. A recent paradigm put forth by Boyle $\textit{et al.}$ (CCS 2018, Crypto 2019) showed how pseudorandom correlation generators (PCG) can be used to generate large amounts of useful forms of correlated (pseudo)randomness, using minimal interactions followed solely by local computations, yielding silent secure two-party computation protocols (protocols where the preprocessing phase requires almost no communication). An additional property called programmability allows to extend this to build N-party protocols. However, known constructions for programmable PCG's can only produce OLE's over large fields, and use rather new splittable Ring-LPN assumption. In this work, we overcome both limitations. To this end, we introduce the quasi-abelian syndrome decoding problem (QA-SD), a family of assumptions which generalises the well-established quasi-cyclic syndrome decoding assumption. Building upon QA-SD, we construct new programmable PCG's for OLE's over any field $\mathbb{F}_q$ with $q>2$. Our analysis also sheds light on the security of the ring-LPN assumption used in Boyle $\textit{et al.}$ (Crypto 2020). Using our new PCG's, we obtain the first efficient N-party silent secure computation protocols for computing general arithmetic circuit over $\mathbb{F}_q$ for any $q>2$.Comment: This is a long version of a paper accepted at CRYPTO'2