MathCheck2: A SAT+CAS Verifier for Combinatorial Conjectures

Bright, Curtis; Heinle, Albert; Kotsireas, Ilias S.; Nejati, Saeed; Czarnecki, Krzysztof

doi:10.1007/978-3-319-45641-6_9

Cited by 17 publications

(13 citation statements)

References 26 publications

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“…Historically, less attention was paid to the even order cases, although generalizations of Williamson matrices were explicitly constructed in even orders by Wallis (1974) as well as Agayan and Sarukhanyan (1981). Williamson matrices were constructed in all even orders up to 22 by Kotsireas and Koukouvinos (2006), up to 34 by Bright et al (2016), and up to 42 by Zulkoski et al (2017). Kotsireas and Koukouvinos (2006) provided a exhaustive search up to order 18 but otherwise these works did not contain a complete enumerations.…”

Section: The Williamson Conjecturementioning

confidence: 99%

“…As an example of this, consider the case of searching for Williamson sequences using a SAT solver. One could encode Definition 5 in CNF format by using Boolean variables to represent the entries in the Williamson sequences and by using binary adders to encode the summations; such a method was used by Bright et al (2016). However, one could also use the equivalent definition given in Theorem 7.…”

Section: Programmatic Sat and Smtmentioning

confidence: 99%

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Applying computer algebra systems with SAT solvers to the Williamson conjecture

Bright

Kotsireas

2020

Journal of Symbolic Computation

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We employ tools from the fields of symbolic computation and satisfiability checkingnamely, computer algebra systems and SAT solvers-to study the Williamson conjecture from combinatorial design theory and increase the bounds to which Williamson matrices have been enumerated. In particular, we completely enumerate all Williamson matrices of even order up to and including 70 which gives us deeper insight into the behaviour and distribution of Williamson matrices. We find that, in contrast to the case when the order is odd, Williamson matrices of even order are quite plentiful and exist in every even order up to and including 70. As a consequence of this and a new construction for 8-Williamson matrices we construct 8-Williamson matrices in all odd orders up to and including 35. We additionally enumerate all Williamson matrices whose orders are divisible by 3 and less than 70, finding one previously unknown set of Williamson matrices of order 63.

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Section: The Williamson Conjecturementioning

confidence: 99%

Section: Programmatic Sat and Smtmentioning

confidence: 99%

Applying computer algebra systems with SAT solvers to the Williamson conjecture

Bright

Kotsireas

2020

Journal of Symbolic Computation

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“…The developer thus has more finegrained control over the power of the SAT solver. This architecture has also shown to be useful in solving problems in combinatorics [7], and much more effective than only using a normal CNF encoding. Figure 1 shows the block diagram of a CDCL SAT solver and the connection of programmatic components (shaded blocks) to the main components.…”

Section: Programmatic Interface In Sat Solversmentioning

confidence: 99%

CDCL(Crypto) SAT Solvers for Cryptanalysis

Nejati¹

2020

Preprint

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Over the last two decades, we have seen a dramatic improvement in the efficiency of conflict-driven clause-learning Boolean satisfiability (CDCL SAT) solvers on industrial problems from a variety of domains. The availability of such powerful general-purpose search tools as SAT solvers has led many researchers to propose SATbased methods for cryptanalysis, including techniques for finding collisions in hash functions and breaking symmetric encryption schemes. Most of the previously proposed SAT-based cryptanalysis approaches are blackbox techniques, in the sense that the cryptanalysis problem is encoded as a SAT instance and then a CDCL SAT solver is invoked to solve the said instance. A weakness of this approach is that the encoding thus generated may be too large for any modern solver to solve efficiently. Perhaps a more important weakness of this approach is that the solver is in no way specialized or tuned to solve the given instance. To address these issues, we propose an approach called CDCL(Crypto) (inspired by the CDCL(T ) paradigm in Satisfiability Modulo Theory solvers) to tailor the internal subroutines of the CDCL SAT solver with domain-specific knowledge about cryptographic primitives. Specifically, we extend the propagation and conflict analysis subroutines of CDCL solvers with specialized codes that have knowledge about the cryptographic primitive being analyzed by the solver. We demonstrate the power of this approach in the differential path and algebraic fault analysis of hash functions. Our initial results are very encouraging and reinforce the notion that this approach is a significant improvement over blackbox SAT-based cryptanalysis.

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“…These searches discovered that Williamson matrices don't exist in the orders n = 35, 47, 53, and 59, but exist in all other orders that were searched. Using MathCheck we were able to provide exhaustive searches for all orders n ≤ 70 divisible by 2 or 3 (finding over 100,000 new sets of Williamson matrices) [13,12] and verified the counterexample n = 35 [10].…”

Section: The Sat+cas Paradigmmentioning

confidence: 99%

The SAT+CAS Method for Combinatorial Search with Applications to Best Matrices

Bright

Djoković

Kotsireas

2019

EasyChair Preprints

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In this paper, we provide an overview of the SAT+CAS method that combines satisfiability checkers (SAT solvers) and computer algebra systems (CAS) to resolve combinatorial conjectures, and present new results vis-à-vis best matrices. The SAT+CAS method is a variant of the Davis-Putnam-Logemann-Loveland DPLL(T ) architecture, where the T solver is replaced by a CAS. We describe how the SAT+CAS method has been previously used to resolve many open problems from graph theory, combinatorial design theory, and number theory, showing that the method has broad applications across a variety of fields. Additionally, we apply the method to construct the largest best matrices yet known and present new skew Hadamard matrices constructed from best matrices. We show the best matrix conjecture (that best matrices exist in all orders of the form r 2 + r + 1) which was previously known to hold for r ≤ 6 also holds for r = 7. We also confirmed the results of the exhaustive searches that have been previously completed for r ≤ 6.

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MathCheck2: A SAT+CAS Verifier for Combinatorial Conjectures

Cited by 17 publications

References 26 publications

Applying computer algebra systems with SAT solvers to the Williamson conjecture

Applying computer algebra systems with SAT solvers to the Williamson conjecture

CDCL(Crypto) SAT Solvers for Cryptanalysis

The SAT+CAS Method for Combinatorial Search with Applications to Best Matrices

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