Linearity Measures for MQ Cryptography

We propose a new general framework for the security of multivariate quadratic (\mathcal{MQ}) schemes with respect to attacks that exploit the existence of linear subspaces. We adopt linearity measures that have been used traditionally to estimate the security of symmetric cryptographic primitives, namely the nonlinearity measure for vectorial functions introduced by Nyberg at Eurocrypt \u2792, and the $(s, t)$--linearity measure introduced recently by Boura and Canteaut at FSE\u2713. We redefine some properties of \mathcal{MQ} cryptosystems in terms of these known symmetric cryptography notions, and show that our new framework is a compact generalization of several known attacks in \mathcal{MQ} cryptography against single field schemes. We use the framework to explain various pitfalls regarding the successfulness of these attacks. Finally, we argue that linearity can be used as a solid measure for the susceptibility of \mathcal{MQ} schemes to these attacks, and also as a necessary tool for prudent design practice in \mathcal{MQ} cryptography

Kryptowochenende 2006 - Workshop über Kryptographie

Das Kryptowochenende ist eine Aktivität der Fachgruppe Angewandte Kryptologie in der Gesellschaft für Informatik (GI) mit dem Ziel, Nachwuchswissenschaftlern, etablierten Forschern und Praktikern auf dem Gebiet der Kryptologie und Computersicherheit die Möglichkeit zu bieten, Kontakte über die eigene Universität hinaus zu knüpfen und sich mit Kollegen aus dem Fachgebiet auszutauschen. Die Vorträge decken ein breites Spektrum ab, von noch laufenden Projekten bis zu abgeschlossenen Forschungsarbeiten, die zeitnah auch auf Konferenzen publiziert wurden bzw. werden sollen. Das erste Kryptowochenende hat stattgefunden vom 01.-02. Juli 2006 im Tagungszentrum der Universität Mannheim im Kloster Bronnbach. Die Beiträge zu diesem Workshop sind im vorliegenden Tagungsband zusammengefasst

A Polynomial-Time Key-Recovery Attack on MQQ Cryptosystems

International audienceWe investigate the security of the family of MQQ public key cryptosystems using multivariate quadratic quasigroups (MQQ). These cryptosystems show especially good performance properties. In particular, the MQQ-SIG signature scheme is the fastest scheme in the ECRYPT benchmarking of cryptographic systems (eBACS). We show that both the signature scheme MQQ-SIG and the encryption scheme MQQ-ENC, although using different types of MQQs, share a common algebraic structure that introduces a weakness in both schemes. We use this weakness to mount a successful polynomial time key-recovery attack. Our key-recovery attack finds an equivalent key using the idea of so-called {\it good keys} that reveals the structure gradually. In the process we need to solve a MinRank problem that, because of the structure, can be solved in polynomial-time assuming some mild algebraic assumptions. We highlight that our theoretical results work in characteristic $2$ which is known to be the most difficult case to address in theory for MinRank attacks. Also, we emphasize that our attack works without any restriction on the number of polynomials removed from the public-key, that is, using the minus modifier. This was not the case for previous MinRank like-attacks against \MQ\ schemes. From a practical point of view, we are able to break an MQQ-SIG instance of $80$ bits security in less than $2$ days, and one of the more conservative MQQ-ENC instances of $128$ bits security in little bit over $9$ days. Altogether, our attack shows that it is very hard to design a secure public key scheme based on an easily invertible MQQ structure

Equivalent Keys in Multivariate Quadratic Public Key Systems

Multivariate Quadratic public key schemes have been suggested back in 1985 by Matsumoto and Imai as an alternative for the RSA scheme. Since then, several other schemes have been proposed, for example Hidden Field Equations, Unbalanced Oil and Vinegar schemes, and Stepwise Triangular Schemes. All these schemes have a rather large key space for a secure choice of parameters. Surprisingly, the question of equivalent keys has not been discussed in the open literature until recently. In this article, we show that for all basic classes mentioned above, it is possible to reduce the private --- and hence the public --- key space by several orders of magnitude. For the Matsumoto-Imai scheme, we are even able to show that the reductions we found are the only ones possible, i.e., that these reductions are tight. While the theorems developed in this article are of independent interest themselves as they broaden our understanding of Multivariate Quadratic public key systems, we see applications of our results both in cryptanalysis and in memory efficient implementations of MQ-schemes

New Directions in Multivariate Public Key Cryptography

Most public key cryptosystems used in practice are based on integer factorization or discrete logarithms (in finite fields or elliptic curves). However, these systems suffer from two potential drawbacks. First, they must use large keys to maintain security, resulting in decreased efficiency. Second, if large enough quantum computers can be built, Shor\u27s algorithm will render them completely insecure. Multivariate public key cryptosystems (MPKC) are one possible alternative. MPKC makes use of the fact that solving multivariate polynomial systems over a finite field is an NP-complete problem, for which it is not known whether there is a polynomial algorithm on quantum computers. The main goal of this work is to show how to use new mathematical structures, specifically polynomial identities from algebraic geometry, to construct new multivariate public key cryptosystems. We begin with a basic overview of MPKC and present several significant cryptosystems that have been proposed. We also examine in detail some of the most powerful attacks against MPKCs. We propose a new framework for constructing multivariate public key cryptosystems and consider several strategies for constructing polynomial identities that can be utilized by the framework. In particular, we have discovered several new families of polynomial identities. Finally, we propose our new cryptosystem and give parameters for which it is secure against known attacks on MPKCs

Public Key Block Cipher Based on Multivariate Quadratic Quasigroups

We have designed a new class of public key algorithms based on quasigroup string transformations using a specific class of quasigroups called \emph{multivariate quadratic quasigroups (MQQ)}. Our public key algorithm is a bijective mapping, it does not perform message expansions and can be used both for encryption and signatures. The public key consist of $n$ quadratic polynomials with $n$ variables where $n=140, 160, \ldots$. A particular characteristic of our public key algorithm is that it is very fast and highly parallelizable. More concretely, it has the speed of a typical modern symmetric block cipher -- the reason for the phrase \emph{ A Public Key Block Cipher } in the title of this paper. Namely the reference C code for the 160--bit variant of the algorithm performs decryption in less than 11,000 cycles (on Intel Core 2 Duo -- using only one processor core), and around 6,000 cycles using two CPU cores and OpenMP 2.0 library. However, implemented in Xilinx Virtex-5 FPGA that is running on 249.4 MHz it achieves decryption throughput of 399 Mbps, and implemented on four Xilinx Virtex-5 chips that are running on 276.7 MHz it achieves encryption throughput of 44.27 Gbps. Compared to fastest RSA implementations on similar FPGA platforms, MQQ algorithm is more than 10,000 times faster

A method of Weil sum in multivariate quadratic cryptosystem

A new cryptanalytic application is proposed for a number theoretic tool Weil sum to the birthday attack against multivariate quadratic trapdoor function. This new customization of the birthday attack is developed by evaluating the explicit Weil sum of the underlying univariate polynomial and the exact number of solutions of the associated bivariate equation. I designed and implemented new algorithms for computing Weil sum values so that I could explicitly identify some class of weak Dembowski- Ostrom polynomials and the equivalent forms in the multivariate quadratic trapdoor function. This customized attack, also regarded as an equation solving algorithm for the system of some special quadratic equations over finite fields, is fundamentally different from the Grobner basis methods. The theoretical observations and experiments show that the required computational complexity of the attack on these weak polynomial instances can be asymptotically less than the square root complexity of the common birthday attack by a factor as large as 2^(n/8) in terms of the extension degree n of F2n. I also suggest a few open problems that any MQ-based short signature scheme must explicitly take into account for the basic design principles

Average Case and Distributional Analysis of Dual-Pivot Quicksort

In 2009, Oracle replaced the long-serving sorting algorithm in its Java 7 runtime library by a new dual-pivot Quicksort variant due to Vladimir Yaroslavskiy. The decision was based on the strikingly good performance of Yaroslavskiy's implementation in running time experiments. At that time, no precise investigations of the algorithm were available to explain its superior performance—on the contrary: previous theoretical studies of other dual-pivot Quicksort variants even discouraged the use of two pivots. In 2012, two of the authors gave an average case analysis of a simplified version of Yaroslavskiy's algorithm, proving that savings in the number of comparisons are possible. However, Yaroslavskiy's algorithm needs more swaps, which renders the analysis inconclusive. To force the issue, we herein extend our analysis to the fully detailed style of Knuth: we determine the exact number of executed Java Bytecode instructions. Surprisingly, Yaroslavskiy's algorithm needs sightly more Bytecode instructions than a simple implementation of classic Quicksort—contradicting observed running times. As in Oracle's library implementation, we incorporate the use of Insertionsort on small subproblems and show that it indeed speeds up Yaroslavskiy's Quicksort in terms of Bytecodes; but even with optimal Insertionsort thresholds, the new Quicksort variant needs slightly more Bytecode instructions on average. Finally, we show that the (suitably normalized) costs of Yaroslavskiy's algorithm converge to a random variable whose distribution is characterized by a fixed-point equation. From that, we compute variances of costs and show that for large n, costs are concentrated around their mean