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

CTC2 and Fast Algebraic Attacks on Block Ciphers Revisited

The cipher CTC (Courtois Toy Cipher) has been designed to demonstrate that it is possible to break on a PC a block cipher with good diffusion and very small number of known (or chosen) plaintexts. It has however never been designed to withstand all known attacks on block ciphers and Dunkelman and Keller have shown that a few bits of the key can be recovered by Linear Cryptanalysis (LC) - which cannot however compromise the security of a large key. This weakness can easily be avoided: in this paper we give a specification of CTC2, a tweaked version of CTC. The new cipher is MUCH more secure than CTC against LC and the key scheduling of CTC has been extended to use any key size, independently from the block size. Otherwise, there is little difference between CTC and CTC2. We will show that up to 10 rounds of CTC2 can be broken by simple algebraic attacks

On Splitting a Point with Summation Polynomials in Binary Elliptic Curves

Recent research for efficient algorithms for solving the discrete logarithm (DL) problem on elliptic curves depends on the difficult question of the feasibility of index calculus which would consist of splitting EC points into sums of points lying in a certain subspace. A natural algebraic approach towards this goal is through solving systems of non linear multivariate equations derived from the so called summation polynomials which method have been proposed by Semaev in 2004 [12]. In this paper we consider simplified variants of this problem with splitting in two or three parts in binary curves. We propose three algorithms with running time of the order of 2^n/3 for both problems. It is not clear how to interpret these results but they do in some sense violate the generic group model for these curves

Cube Attack on Courtois Toy Cipher

Abstract. The cube attack has been introduced by Itai Dinur and Adi Shamir [8] as a known plaintext attack on symmetric primitives. The attack has been applied to reduced variants of the stream ciphers Trivium [3, 8] and Grain-128 [2], reduced to three rounds variant of the block cipher Serpent [9] and reduced version of the hash function MD6 [3]. In the special case the attack has appeared in the M. Vielhaber ePrint articles [13, 14], where it has been named AIDA (Algebraic Initial Value Differential Attack ) and applied to the modified versions of Trivium. In this paper, we present the experimental results of application the cube attack to four rounds of the Courtois Toy Cipher (CTC) with the full recovery of 120-bit key. After that we extend the attack to five rounds by applying the meet-in-the-middle principle. Key words: Cube attack, symmetric primitives, Boolean polynomials, CTC, the meet-in-the-middle metho

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

D.STVL.7 - Algebraic cryptanalysis of symmetric primitives

The recent development of algebraic attacks can be considered an important breakthrough in the analysis of symmetric primitives; these are powerful techniques that apply to both block and stream ciphers (and potentially hash functions). The basic principle of these techniques goes back to Shannon's work: they consist in expressing the whole cryptographic algorithm as a large system of multivariate algebraic equations (typically over F2), which can be solved to recover the secret key. Efficient algorithms for solving such algebraic systems are therefore the essential ingredients of algebraic attacks. Algebraic cryptanalysis against symmetric primitives has recently received much attention from the cryptographic community, particularly after it was proposed against some LFSR- based stream ciphers and against the AES and Serpent block ciphers. This is currently a very active area of research. In this report we discuss the basic principles of algebraic cryptanalysis of stream ciphers and block ciphers, and review the latest developments in the field. We give an overview of the construction of such attacks against both types of primitives, and recall the main algorithms for solving algebraic systems. Finally we discuss future research directions

How Fast can be Algebraic Attacks on Block Ciphers?

In this paper we give a specification of a new block cipher that can be called the Courtois Toy Cipher (CTC). It is quite simple, and yet very much like any other known block cipher. If the parameters are large enough, it should evidently be secure against all known attack methods.However, we are not proposing a new method for encrypting sensitive data, but rather a research tool that should allow us (and other researchers) to experiment with algebraic attacks on block ciphers and obtain interesting results using a PC with reasonable quantity of RAM. For this reason the S-box of this cipher has only 3-bits, which is quite small. Ciphers wit

High Saturation Complete Graph Approach for EC Point Decomposition and ECDL Problem

One of the key questions in contemporary applied cryptography is whether there exist an efficient algorithm for solving the discrete logarithm problem in elliptic curves. The primary approach for this problem is to try to solve a certain system of polynomial equations. Current attempts try to solve them directly with existing software tools which does not work well due to their very loosely connected topology and illusory reliance on degree falls. A deeper reflection on what makes systems of algebraic equations efficiently solvable is missing. In this paper we propose a new approach for solving this type of polynomial systems which is radically different than current approaches. We carefully engineer systems of equations with excessively dense topology obtained from a complete clique/biclique graphs and hypergraphs and unique special characteristics. We construct a sequence of systems of equations with a parameter K and argue that asymptotically when K grows the system of equations achieves a high level of saturation with lim_{K\to\infty} F/T = 1 which allows to reduce the regularity degree and makes that polynomial equations over finite fields may become efficiently solvable

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