Block ciphers sensitive to Groebner Basis Attacks

We construct and analyze Feistel and SPN ciphers that have a sound design strategy against linear and differential attacks but for which the encryption process can be described by very simple polynomial equations. For a block and key size of 128 bits, we present ciphers for which practical Groebner basis attacks can recover the full cipher key requiring only a minimal number of plaintext/ciphertext pairs. We show how Groebner bases for a subset of these ciphers can be constructed with neglegible computational effort. This reduces the key recovery problem to a Groebner basis conversion problem. By bounding the running time of a Groebner basis conversion algorithm, FGLM, we demonstrate the existence of block ciphers resistant against differential and linear cryptanalysis but vulnerable against Groebner basis attacks

Security Evaluation of MISTY Structure with SPN Round Function

This paper deals with the security of MISTY structure with SPN round function. We study the lower bound of the number of active s-boxes for differential and linear characteristics of such block cipher construction. Previous result shows that the differential bound is consistent with the case of Feistel structure with SPN round function, yet the situation changes when considering the linear bound. We carefully revisit such issue, and prove that the same bound in fact could be obtained for linear characteristic. This result combined with the previous one thus demonstrates a similar practical secure level for both Feistel and MISTY structures. Besides, we also discuss the resistance of MISTY structure with SPN round function against other kinds of cryptanalytic approaches including the integral cryptanalysis and impossible differential cryptanalysis. We confirm the existence of 6-round integral distinguishers when the linear transformation of the round function employs a binary matrix (i.e., the element in the matrix is either 0 or 1), and briefly describe how to characterize 5/6/7-round impossible differentials through the matrix-based method

Block Ciphers - Focus On The Linear Layer (feat. PRIDE): Full Version

The linear layer is a core component in any substitution-permutation network block cipher. Its design significantly influences both the security and the efficiency of the resulting block cipher. Surprisingly, not many general constructions are known that allow to choose trade-offs between security and efficiency. Especially, when compared to Sboxes, it seems that the linear layer is crucially understudied. In this paper, we propose a general methodology to construct good, sometimes optimal, linear layers allowing for a large variety of trade-offs. We give several instances of our construction and on top underline its value by presenting a new block cipher. PRIDE is optimized for 8-bit micro-controllers and significantly outperforms all academic solutions both in terms of code size and cycle count

On the Design of Bit Permutation Based Ciphers - The Interplay Among S-box, Bit Permutation and Key-addition

Bit permutation based block ciphers, like PRESENT and GIFT, are well-known for their extreme lightweightness in hardware implementation. However, designing such ciphers comes with one major challenge - to ensure strong cryptographic properties simply depending on the combination of three components, namely S-box, a bit permutation and a key addition function. Having a wrong combination of components could lead to weaknesses. In this article, we studied the interaction between these components, improved the theoretical security bound of GIFT and highlighted the potential pitfalls associated with a bit permutation based primitive design. We also conducted analysis on TRIFLE, a first-round candidate for the NIST lightweight cryptography competition, where our findings influenced the elimination of TRIFLE from second-round of the NIST competition. In particular, we showed that internal state bits of TRIFLE can be partially decrypted for a few rounds even without any knowledge of the key

Application of Fault Analysis to Some Cryptographic Standards

Cryptanalysis methods can be classified as pure mathematical attacks, such as linear and differential cryptanalysis, and implementation dependent attacks such as power analysis and fault analysis. Pure mathematical attacks exploit the mathematical structure of the cipher to reveal the secret key inside the cipher. On the other hand, implementation dependent attacks assume that the attacker has access to the cryptographic device to launch the attack. Fault analysis is an example of a side channel attack in which the attacker is assumed to be able to induce faults in the cryptographic device and observe the faulty output. Then, the attacker tries to recover the secret key by combining the information obtained from the faulty and the correct outputs. Even though fault analysis attacks may require access to some specialized equipment to be able to insert faults at specific locations or at specific times during the computation, the resulting attacks usually have time and memory complexities which are far more practical as compared to pure mathematical attacks. Recently, several AES-based primitives were approved as new cryptographic standards throughout the world. For example, Kuznyechik was approved as the standard block cipher in Russian Federation, and Kalyna and Kupyna were approved as the standard block cipher and the hash function, respectively, in Ukraine. Given the importance of these three new primitives, in this thesis, we analyze their resistance against fault analysis attacks. Firstly, we modified a differential fault analysis (DFA) attack that was applied on AES and applied it on Kuzneychik. Application of DFA on Kuznyechik was not a trivial task because of the linear transformation layer used in the last round of Kuznyechik. In order to bypass the effect of this linear transformation operation, we had to use an equivalent representation of the last round which allowed us to recover the last two round keys using a total of four faults and break the cipher. Secondly, we modified the attack we applied on Kuzneychik and applied it on Kalyna. Kalyna has a complicated key scheduling and it uses modulo 264 addition operation for applying the first and last round keys. This makes Kalyna more resistant to DFA as com- pared to AES and Kuznyechik but it is still practically breakable because the number of key candidates that can be recovered by DFA can be brute-forced in a reasonable time. We also considered the case where the SBox entries of Kalyna are not known and showed how to recover a set of candidates for the SBox entries. Lastly, we applied two fault analysis attacks on Kupyna hash function. In the first case, we assumed that the SBoxes and all the other function parameters are known, and in the second case we assumed that the SBoxes were kept secret and attacked the hash function accordingly. Kupyna can be used as the underlying hash function for the construction of MAC schemes such as secret IV, secret prefix, HMAC or NMAC. In our analysis, we showed that secret inputs of Kupyna can be recovered using fault analysis. To conclude, we analyzed two newly accepted standard ciphers (Kuznyechik, Kalyna) and one newly approved standard hash function (Kupyna) for their resistance against fault attacks. We also analyzed Kalyna and Kupyna with the assumption that these ciphers can be deployed with secret user defined SBoxes in order to increase their security

C-DIFFERENTIALS AND GENERALIZED CRYPTOGRAPHIC PROPERTIES OF VECTORIAL BOOLEAN AND P-ARY FUNCTIONS

This dissertation investigates a newly defined cryptographic differential, called a c-differential, and its relevance to the nonlinear substitution boxes of modern symmetric block ciphers. We generalize the notions of perfect nonlinearity, bentness, and avalanche characteristics of vectorial Boolean and p-ary functions using the c-derivative and a new autocorrelation function, while capturing the original definitions as special cases (i.e., when c=1). We investigate the c-differential uniformity property of the inverse function over finite fields under several extended affine transformations. We demonstrate that c-differential properties do not hold in general across equivalence classes typically used in Boolean function analysis, and in some cases change significantly under slight perturbations. Thus, choosing certain affine equivalent functions that are easy to implement in hardware or software without checking their c-differential properties could potentially expose an encryption scheme to risk if a c-differential attack method is ever realized. We also extend the c-derivative and c-differential uniformity into higher order, investigate some of their properties, and analyze the behavior of the inverse function's second order c-differential uniformity. Finally, we analyze the substitution boxes of some recognizable ciphers along with certain extended affine equivalent variations and document their performance under c-differential uniformity.Commander, United States NavyApproved for public release. Distribution is unlimited

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

Cryptanalysis of ARX-based White-box Implementations

At CRYPTO’22, Ranea, Vandersmissen, and Preneel proposed a new way to design white-box implementations of ARX-based ciphers using so-called implicit functions and quadratic-affine encodings. They suggest the Speck block-cipher as an example target. In this work, we describe practical attacks on the construction. For the implementation without one of the external encodings, we describe a simple algebraic key recovery attack. If both external encodings are used (the main scenario suggested by the authors), we propose optimization and inversion attacks, followed by our main result - a multiple-step round decomposition attack and a decomposition-based key recovery attack. Our attacks only use the white-box round functions as oracles and do not rely on their description. We implemented and verified experimentally attacks on white-box instances of Speck-32/64 and Speck-64/128. We conclude that a single ARX-round is too weak to be used as a white-box round