Predicting Outcomes of ElimLin Attack on Lightweight Block Cipher Simon

There are two major families in cryptanalytic attacks on symmetric ciphers: statistical attacks and algebraic attacks. In this position paper we argue that algebraic cryptanalysis has not yet been developed properly due to the weakness of the theory which has substantial difficulty to prove most basic results on the number of linearly independent equations in algebraic attacks. Consequently most authors present a restricted range of attacks which are shown experimentally to work with their computer but refrain from claiming results which would work on a larger computer but have not yet been tested. For example in recent 2015 work of Raddum we discover that (experimentally) ElimLin attack breaks up to 16 rounds of Simon block cipher however it is hard to know what happens for 17 rounds. In this paper we argue that one CAN predict and model the behavior of such attacks and evaluate complexity of the attacks which we cannot yet execute. To the best of our knowledge this has never been done before

ElimLin Algorithm Revisited

ElimLin is a simple algorithm for solving polynomial systems of multivariate equations over small finite fields. It was initially proposed as a single tool by Courtois to attack DES. It can reveal some hidden linear equations existing in the ideal generated by the system. We report a number of key theorems on ElimLin. Our main result is to characterize ElimLin in terms of a sequence of intersections of vector spaces. It implies that the linear space generated by ElimLin is invariant with respect to any variable ordering during elimination and substitution. This can be seen as surprising given the fact that it eliminates variables. On the contrary, monomial ordering is a crucial factor in Gröbner basis algorithms such as F4. Moreover, we prove that the result of ElimLin is invariant with respect to any affine bijective variable change. Analyzing an overdefined dense system of equations, we argue that to obtain more linear equations in the succeeding iteration in ElimLin some restrictions should be satisfied. Finally, we compare the security of LBlock and MIBS block ciphers with respect to algebraic attacks and propose several attacks on Courtois Toy Cipher version 2 (CTC2) with distinct parameters using ElimLin

Two philosophies for solving non-linear equations in algebraic cryptanalysis

Algebraic Cryptanalysis [45] is concerned with solving of particular systems of multivariate non-linear equations which occur in cryptanalysis. Many different methods for solving such problems have been proposed in cryptanalytic literature: XL and XSL method, Gröbner bases, SAT solvers, as well as many other. In this paper we survey these methods and point out that the main working principle in all of them is essentially the same. One quantity grows faster than another quantity which leads to a “phase transition” and the problem becomes efficiently solvable. We illustrate this with examples from both symmetric and asymmetric cryptanalysis. In this paper we point out that there exists a second (more) general way of formulating algebraic attacks through dedicated coding techniques which involve redundancy with addition of new variables. This opens numerous new possibilities for the attackers and leads to interesting optimization problems where the existence of interesting equations may be somewhat deliberately engineered by the attacker

On Selection of Samples in Algebraic Attacks and a New Technique to Find Hidden Low Degree Equations

The best way of selecting samples in algebraic attacks against block ciphers is not well explored and understood. We introduce a simple strategy for selecting the plaintexts and demonstrate its strength by breaking reduced-round KATAN32 and LBlock. In both cases, we present a practical attack which outperforms previous attempts of algebraic cryptanalysis whose complexities were close to exhaustive search. The attack is based on the selection of samples using cube attack and ElimLin which was presented at FSE’12, and a new technique called Universal Proning. In the case of LBlock, we break 10 out of 32 rounds. In KATAN32, we break 78 out of 254 rounds. Unlike previous attempts which break smaller number of rounds, we do not guess any bit of the key and we only use structural properties of the cipher to be able to break a higher number of rounds with much lower complexity. We show that cube attacks owe their success to the same properties and therefore, can be used as a heuristic for selecting the samples in an algebraic attack. The performance of ElimLin is further enhanced by the new Universal Proning technique, which allows to discover linear equations that are not found by ElimLin

Algebraic Cryptanalysis of Deterministic Symmetric Encryption

Deterministic symmetric encryption is widely used in many cryptographic applications. The security of deterministic block and stream ciphers is evaluated using cryptanalysis. Cryptanalysis is divided into two main categories: statistical cryptanalysis and algebraic cryptanalysis. Statistical cryptanalysis is a powerful tool for evaluating the security but it often requires a large number of plaintext/ciphertext pairs which is not always available in real life scenario. Algebraic cryptanalysis requires a smaller number of plaintext/ciphertext pairs but the attacks are often underestimated compared to statistical methods. In algebraic cryptanalysis, we consider a polynomial system representing the cipher and a solution of this system reveals the secret key used in the encryption. The contribution of this thesis is twofold. Firstly, we evaluate the performance of existing algebraic techniques with respect to number of plaintext/ciphertext pairs and their selection. We introduce a new strategy for selection of samples. We build this strategy based on cube attacks, which is a well-known technique in algebraic cryptanalysis. We use cube attacks as a fast heuristic to determine sets of plaintexts for which standard algebraic methods, such as Groebner basis techniques or SAT solvers, are more efficient. Secondly, we develop a~new technique for algebraic cryptanalysis which allows us to speed-up existing Groebner basis techniques. This is achieved by efficient finding special polynomials called mutants. Using these mutants in Groebner basis computations and SAT solvers reduces the computational cost to solve the system. Hence, both our methods are designed as tools for building polynomial system representing a cipher. Both tools can be combined and they lead to a significant speedup, even for very simple algebraic solvers

On Selection of Samples in Algebraic Attacks and a New Technique to Find Hidden Low Degree Equations

The best way of selecting samples in algebraic attacks against block ciphers is not well explored and understood. We introduce a simple strategy for selecting the plaintexts and demonstrate its strength by breaking reduced-round KATAN, LBLOCK and SIMON. For each case, we present a practical attack on reduced round version which outperforms previous attempts of algebraic cryptanalysis whose complexities were close to exhaustive search. The attack is based on the selection of samples using cube attack and ELIMLIN which was presented at FSE'12, and a new technique called proning. In the case of LBLOCK, we break 10 out of 32 rounds. In KATAN, we break 78 out of 254 rounds. Unlike previous attempts which break smaller number of rounds, we do not guess any bit of the key and we only use structural properties of the cipher to be able to break a higher number of rounds with much lower complexity. We show that cube attacks owe their success to the same properties and therefore, can be used as a heuristic for selecting the samples in an algebraic attack. The performance of ELIMLIN is further enhanced by the new proning technique, which allows to discover linear equations that are not found by ELIMLIN

Optimization and Guess-then-Solve Attacks in Cryptanalysis

In this thesis we study two major topics in cryptanalysis and optimization: software algebraic cryptanalysis and elliptic curve optimizations in cryptanalysis. The idea of algebraic cryptanalysis is to model a cipher by a Multivariate Quadratic (MQ) equation system. Solving MQ is an NP-hard problem. However, NP-hard problems have a point of phase transition where the problems become easy to solve. This thesis explores different optimizations to make solving algebraic cryptanalysis problems easier. We first worked on guessing a well-chosen number of key bits, a specific optimization problem leading to guess-then-solve attacks on GOST cipher. In addition to attacks, we propose two new security metrics of contradiction immunity and SAT immunity applicable to any cipher. These optimizations play a pivotal role in recent highly competitive results on full GOST. This and another cipher Simon, which we cryptanalyzed were submitted to ISO to become a global encryption standard which is the reason why we study the security of these ciphers in a lot of detail. Another optimization direction is to use well-selected data in conjunction with Plaintext/Ciphertext pairs following a truncated differential property. These allow to supplement an algebraic attack with extra equations and reduce solving time. This was a key innovation in our algebraic cryptanalysis work on NSA block cipher Simon and we could break up to 10 rounds of Simon64/128. The second major direction in our work is to inspect, analyse and predict the behaviour of ElimLin attack the complexity of which is very poorly understood, at a level of detail never seen before. Our aim is to extrapolate and discover the limits of such attacks, and go beyond with several types of concrete improvement. Finally, we have studied some optimization problems in elliptic curves which also deal with polynomial arithmetic over finite fields. We have studied existing implementations of the secp256k1 elliptic curve which is used in many popular cryptocurrency systems such as Bitcoin and we introduce an optimized attack on Bitcoin brain wallets and improved the state of art attack by 2.5 times

Algebraic Fault Analysis of Katan

This paper presents a new and more realistic model for fault attacks and statistical and algebraic techniques to improve fault analysis in general. Our algebraic techniques is an adapted solver for systems of equations based on ElimLin and XSL. We use these techniques to introduce two new fault attacks on the hardware oriented block cipher Katan32 from the Katan family of block ciphers. We are able to break full Katan using $4$ faults and $2^{29.04}$ Katan evaluations with a theoretical statistical fault attack and $7.19$ faults in $2^{27.2}$ Katan evaluations with a tested algebraic one. This is a great improvement over the existing fault attacks which need $115$ and $140$ faults respectively. Furthermore, our algebraic attack can be executed on a normal computer

Advanced Algebraic Attack on Trivium

This paper presents an algebraic attack against Trivium that breaks 625 rounds using only $4096$ bits of output in an overall time complexity of $2^{42.2}$ Trivium computations. While other attacks can do better in terms of rounds ($799$), this is a practical attack with a very low data usage (down from $2^{40}$ output bits) and low computation time (down from $2^{62}$). From another angle, our attack can be seen as a proof of concept: how far can algebraic attacks can be pushed when several known techniques are combined into one implementation? All attacks have been fully implemented and tested; our figures are therefore not the result of any potentially error-prone extrapolation, but results of practical experiments