MRHS Solver Based on Linear Algebra and Exhaustive Search

We show how to build a binary matrix from the MRHS representation of a symmetric-key cipher. The matrix contains the cipher represented as an equation system and can be used to assess a cipher\u27s resistance against algebraic attacks. We give an algorithm for solving the system and compute its complexity. The complexity is normally close to exhaustive search on the variables representing the user-selected key. Finally, we show that for some variants of LowMC, the joined MRHS matrix representation can be used to speed up regular encryption in addition to exhaustive key search

Using Local Reduction for the Experimental Evaluation of the Cipher Security

Evaluating the strength of block ciphers against algebraic attacks can be difficult. The attack methods often use different metrics, and experiments do not scale well in practice. We propose a methodology that splits the algebraic attack into a polynomial part (local reduction), and an exponential part (guessing), respectively. The evaluator uses instances with known solutions to estimate the complexity of the attacks, and the response to changing parameters of the problem (e.g. the number of rounds). Although the methodology does not provide a positive answer ("the cipher is secure"), it can be used to construct a negative test (reject weak ciphers), or as a tool of qualitative comparison of cipher designs. Potential applications in other areas of computer science are discussed in the concluding parts of the article

Selected Topics in Cryptanalysis of Symmetric Ciphers

It is well established that a symmetric cipher may be described as a system of Boolean polynomials, and that the security of the cipher cannot be better than the difficulty of solving said system. Compressed Right-Hand Side (CRHS) Equations is but one way of describing a symmetric cipher in terms of Boolean polynomials. The first paper of this thesis provides a comprehensive treatment firstly of the relationship between Boolean functions in algebraic normal form, Binary Decision Diagrams and CRHS equations. Secondly, of how CRHS equations may be used to describe certain kinds of symmetric ciphers and how this model may be used to attempt a key-recovery attack. This technique is not left as a theoretical exercise, as the process have been implemented as an open-source project named CryptaPath. To ensure accessibility for researchers unfamiliar with algebraic cryptanalysis, CryptaPath can convert a reference implementation of the target cipher, as specified by a Rust trait, into the CRHS equations model automatically. CRHS equations are not limited to key-recovery attacks, and Paper II explores one such avenue of CRHS equations flexibility. Linear and differential cryptanalysis have long since established their position as two of the most important cryptanalytical attacks, and every new design since must show resistance to both. For some ciphers, like the AES, this resistance can be mathematically proven, but many others are left to heuristic arguments and computer aided proofs. This work is tedious, and most of the tools require good background knowledge of a tool/technique to transform a design to the right input format, with a notable exception in CryptaGraph. CryptaGraph is written in Rust and transforms a reference implementation into CryptaGraphs underlying data structure automatically. Paper II introduces a new way to use CRHS equations to model a symmetric cipher, this time in such a way that linear and differential trail searches are possible. In addition, a new set of operations allowing us to count the number of active S-boxes in a path is presented. Due to CRHS equations effective initial data compression, all possible trails are captured in the initial system description. As is the case with CRHS equations, the crux is the memory consumption. However, this approach also enables the graph of a CRHS equation to be pruned, allowing the memory consumption to be kept at manageable levels. Unfortunately, pruning nodes also means that we will lose valid, incomplete paths, meaning that the hulls found are probably incomplete. On the flip side, all paths, and their corresponding probabilities, found by the tool are guaranteed to be valid trails for the cipher. This theory is also implemented in an extension of CryptaPath, and the name is PathFinder. PathFinder is also able to automatically turn a reference implementation of a cipher into its CRHS equations-based model. As an additional bonus, PathFinder supports the reference implementation specifications specified by CryptaGraph, meaning that the same reference implementation can be used for both CryptaGraph and PathFinder. Paper III shifts focus onto symmetric ciphers designed to be used in conjunction with FHE schemes. Symmetric ciphers designed for this purpose are relatively new and have naturally had a strong focus on reducing the number of multiplications performed. A multiplication is considered expensive on the noise budget of the FHE scheme, while linear operations are viewed as cheap. These ciphers are all assuming that it is possible to find parameters in the various FHE schemes which allow these ciphers to work well in symbiosis with the FHE scheme. Unfortunately, this is not always possible, with the consequence that the decryption process becomes more costly than necessary. Paper III therefore proposes Fasta, a stream cipher which has its parameters and linear layer especially chosen to allow efficient implementation over the BGV scheme, particularly as implemented in the HElib library. The linear layers are drawn from a family of rotation-based linear transformations, as cyclic rotations are cheap to do in FHE schemes that allow packing of multiple plaintext elements in one FHE ciphertext. Fasta follows the same design philosophy as Rasta, and will never use the same linear layer twice under the same key. The result is a stream cipher tailor-made for fast evaluation in HElib. Fasta shows an improvement in throughput of a factor more than 7 when compared to the most efficient implementation of Rasta.Doktorgradsavhandlin

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

Challenges in nonlinear structural dynamics: New optimisation perspectives

nalysis of structural dynamics is of fundamental importance to countless engineering applications. Analyses in both research and industrial settings have traditionally relied on linear or close to linear approximations of the underlying physics. Perhaps the most pervasive framework, modal analysis, has become the default framework for consideration of linear dynamics. Modern hardware and software solutions have placed linear analysis of structural dynamics squarely in the mainstream. However, as demands for stronger and lighter structures increase, and as advanced manufacturing enables more and more intricate geometries, the assumption of linearity is becoming less and less realistic. This thesis envisages three grand challenges for the treatment of nonlinearity in structural dynamics. These are: nonlinear system identification, exact solutions to nonlinear differential equations, and a nonlinear extension to linear modal analysis. Of these challenges, this thesis presents results pertaining to the latter two. The first component of this thesis is the consideration of methods that may yield exact solutions to nonlinear differential equations. Here, the task of finding an exact solution is cast as a heuristic search problem. The structure of the search problem is analysed with a view to motivate methods that are predisposed to finding exact solutions. To this end, a novel methodology, the affine regression tree, is proposed. The novel approach is compared against alternatives from the literature in an expansive benchmark study. Also considered, are nonlinear extensions to linear modal analysis. Historically, several frameworks have been proposed, each of which is able to retain only a subset of the properties of the linear case. It is argued here that retention of the utilities of linear modal analysis should be viewed as the criteria for a practical nonlinear modal decomposition. A promising direction is seen to be the recently-proposed framework of Worden and Green. The approach takes a machine-learning viewpoint that requires statistical independence between the modal coordinates. In this thesis, a robust consideration of the method from several directions is attempted. Results from several analyses demonstrate that statistical-independence and other inductive biases can be sufficient for a meaningful nonlinear modal decomposition, opening the door to a practical, nonlinear extension to modal analysis. The results in this thesis take small but positive steps towards two pressing challenges facing nonlinear structural dynamics. It is hoped that further work will be able to build upon the results presented here to develop a greater understanding and treatment of nonlinearity in structural dynamics and elsewhere