Cryptography from tensor problems

We describe a new proposal for a trap-door one-way function. The new proposal belongs to the "multivariate quadratic" family but the trap-door is different from existing methods, and is simpler

Algebraic Attacks on RAIN and AIM Using Equivalent Representations

Designing novel symmetric-key primitives for advanced protocols like secure multiparty computation (MPC), fully homomorphic encryption (FHE) and zero-knowledge proof systems (ZK), has been an important research topic in recent years. Many such existing primitives adopt quite different design strategies from conventional block ciphers. Notable features include that many of these ciphers are defined over a large finite field, and that a power map is commonly used to construct the nonlinear component due to its efficiency in these applications as well as its strong resistance against the differential and linear cryptanalysis. In this paper, we target the MPC-friendly ciphers AIM and RAIN used for the post-quantum signature schemes AIMer (CCS 2023 and NIST PQC Round 1 Additional Signatures) and Rainier (CCS 2022), respectively. Specifically, we can find equivalent representations of 2-round RAIN and full-round AIM, respectively, which make them vulnerable to either the polynomial method, or the crossbred algorithm, or the fast exhaustive search attack. Consequently, we can break 2-round RAIN with the 128/192/256-bit key in only 2111/2170/2225 bit operations. For full-round AIM with the 128/192/256-bit key, we could break them in 2136.2/2200.7/2265 bit operations, which are equivalent to about 2115/2178/2241 calls of the underlying primitives. In particular, our analysis indicates that AIM does not reach the required security levels by the NIST competition.</p

Algebraic Attacks on RAIN and AIM Using Equivalent Representations

Designing novel symmetric-key primitives for advanced protocols like secure multiparty computation (MPC), fully homomorphic encryption (FHE) and zero-knowledge proof systems (ZK), has been an important research topic in recent years. Many such existing primitives adopt quite different design strategies from conventional block ciphers. Notable features include that many of these ciphers are defined over a large finite field, and that a power map is commonly used to construct the nonlinear component due to its efficiency in these applications as well as its strong resistance against the differential and linear cryptanalysis. In this paper, we target the MPC-friendly ciphers AIM and RAIN used for the post-quantum signature schemes AIMer (CCS 2023 and NIST PQC Round 1 Additional Signatures) and Rainier (CCS 2022), respectively. Specifically, we can find equivalent representations of 2-round RAIN and full-round AIM, respectively, which make them vulnerable to either the polynomial method, or the crossbred algorithm, or the fast exhaustive search attack. Consequently, we can break 2-round RAIN with the 128/192/256-bit key in only 2111/2170/2225 bit operations. For full-round AIM with the 128/192/256-bit key, we could break them in 2136.2/2200.7/2265 bit operations, which are equivalent to about 2115/2178/2241 calls of the underlying primitives. In particular, our analysis indicates that AIM does not reach the required security levels by the NIST competition.</p

Computational and Algebraic Aspects of the Advanced Encryption Standard

Abstract. The new Advanced Encryption Standard (AES) has been recently selected by the US government to replace the old Data Encryption Standard (DES) for protecting sensitive official information. Due to its simplicity and elegant algebraic structure, the choice of the AES algorithm has motivated the study of a new approach to the analysis of block ciphers. While conventional methods of cryptanalysis (e.g. differential and linear cryptanalysis) are usually based on a “statistical ” approach, where an attacker attempts to construct statistical patterns through many interactions of the cipher, the so-called algebraic attacks exploit the intrinsic algebraic structure of a cipher. More specifically, the attacker expresses the encryption transformation as a set of multivariate polynomial equations and attempts to recover the encryption key by solving the system. In this paper we consider a number of algebraic aspects of the AES, and examine a few computational and algebraic techniques that could be used in the cryptanalysis of cipher. We show how one can express the cipher as a very large, though surprisingly simple, system of multivariate quadratic equations over the finite field F 2 8, and consider some approaches that can be used to solve this system

Performance evaluation of eXtended sparse linearization in GF(2) and GF(28)

XSL (eXtended Sparse Linearization) is a recent algebraic attack aimed at the Advanced Encryption Standard. In order to shed some light into the behavior of the algorithm, which is largely unknown, we have studied XSL on equation systems with variables interpreted either as bits or bytes. The algorithm solves byte-systems much faster than it does bit-systems, which promts us to suggest that if a more compact representation of equation systems can be found, such as one where the variables are 8-byte blocks, or even a more generalized form of 8n-byte blocks, it may be possible to increase the speed of XSL dramatically

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

Algebraic Attacks on RAIN and AIM Using Equivalent Representations

Designing novel symmetric-key primitives for advanced protocols like secure multiparty computation (MPC), fully homomorphic encryption (FHE) and zero-knowledge proof systems (ZK), has been an important research topic in recent years. Many such existing primitives adopt quite different design strategies from conventional block ciphers. Notable features include that many of these ciphers are defined over a large finite field, and that a power map is commonly used to construct the nonlinear component due to its efficiency in these applications as well as its strong resistance against the differential and linear cryptanalysis. In this paper, we target the MPC-friendly ciphers AIM and RAIN used for the post-quantum signature schemes AIMer (CCS 2023 and NIST PQC Round 1 Additional Signatures) and Rainier (CCS 2022), respectively. Specifically, we can find equivalent representations of 2-round RAIN and full-round AIM, respectively, which make them vulnerable to either the polynomial method, or the crossbred algorithm, or the fast exhaustive search attack. Consequently, we can break 2-round RAIN with the 128/192/256-bit key in only 2111/2170/2225 bit operations. For full-round AIM with the 128/192/256-bit key, we could break them in 2136.2/2200.7/2265 bit operations, which are equivalent to about 2115/2178/2241 calls of the underlying primitives. In particular, our analysis indicates that AIM does not reach the required security levels by the NIST competition