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## Synthesizing shortest linear straight-line programs over GF(2) using SAT (2010)

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Venue: | In Proc. SAT ’10, volume 6175 of LNCS |

Citations: | 9 - 1 self |

### Citations

673 |
An extensible SAT-solver
- Een, Sörensson
(Show Context)
Citation Context ...f the verification environment AProVE [6] and the Tseitin implementation of SAT4J [7]. For the top matrix, satisfiability for the threshold value kmin = 23 was shown using e.g. the SAT solver MiniSAT =-=[3]-=-, as reported in [5]. The unsatisfiability proof for kmin − 1 = 22 was only achieved later by CryptoMiniSAT [8] (cf. [4]). The corresponding value of kmin for the bottom matrix is currently unknown. I... |

401 |
On the complexity of derivation in propositional calculus,
- Tseitin
- 1970
(Show Context)
Citation Context ... prenex normal form that can be used as input for a SAT solver (by dropping explicit existential quantification, encoding cardinality constraints using [5, 1], and performing Tseitin’s transformation =-=[14]-=-). δ3(ℓ) = ∧ ⎛ ⎝fℓ,i → ∧ ⎞ (ψ(j, i) ↔ aℓ,j) ⎠ ∧ exactly1(fℓ,1, . . . , fℓ,k) 1≤i≤k 1≤j≤n δ = ∃b1,1. . . . ∃bk,n.∃c1,1. . . ....∃ck,k.∃f1,1. . . . ∃fm,k. β1 ∧ ∧ 1≤ℓ≤m δ3(ℓ) For the implementation of δ ... |

185 | Translating pseudo-Boolean constraints into SAT.
- Een, Sörensson
- 2006
(Show Context)
Citation Context ...mplementations it is beneficial to represent propositional formulae not as trees, but as directed acyclic graphs with sharing of common subformulae. This technique is also known as structural hashing =-=[6]-=-. We perform standard Boolean simplifications (e.g., ϕ ∧ 1 = ϕ), we share Boolean junctor applications modulo commutativity and idempotence (where applicable), and we use varyadic ∧ and ∨. In contrast... |

118 | AProVE 1.2: Automatic Termination Proofs in the Dependency Pair Framework.
- Giesl, Schneider-Kamp, et al.
- 2006
(Show Context)
Citation Context .... . , fℓ,k) 1≤i≤k 1≤j≤n δ = ∃b1,1. . . . ∃bk,n.∃c1,1. . . ....∃ck,k.∃f1,1. . . . ∃fm,k. β1 ∧ ∧ 1≤ℓ≤m δ3(ℓ) For the implementation of δ we used the SAT framework in the verification environment AProVE =-=[8]-=- and the Tseitin implementation from SAT4J [12]. 3.1 Size of the Encoding Given a decision problem with an m × n matrix and a natural number k (where w.l.o.g. m ≤ k holds since for m > k, we could jus... |

54 | SAT Solving for Termination Analysis with Polynomial Interpretations.
- Fuhs, Giesl, et al.
- 2007
(Show Context)
Citation Context ...efit from linearity of the operation ⊕ on GF(2), which means that the absolute positiveness criterion for polynomials [10] (a simple technique commonly used in automated termination provers, cf. e.g. =-=[7]-=-) is not only sound, but also complete. Essentially, the idea is that two linear forms compute the same function iff their coefficients are identical. In this way, we can now drop the inputs x1, . . .... |

47 |
The Sat4j library, release 2.2.
- Berre, Parrain
- 2010
(Show Context)
Citation Context ...r program lengths 12 to 35. They were created using our implementation for [5], which makes use of the SAT framework of the verification environment AProVE [6] and the Tseitin implementation of SAT4J =-=[7]-=-. For the top matrix, satisfiability for the threshold value kmin = 23 was shown using e.g. the SAT solver MiniSAT [3], as reported in [5]. The unsatisfiability proof for kmin − 1 = 22 was only achiev... |

45 | Extending SAT Solvers to Cryptographic Problems.
- Soos, Nohl, et al.
- 2009
(Show Context)
Citation Context ...sfiability for the threshold value kmin = 23 was shown using e.g. the SAT solver MiniSAT [3], as reported in [5]. The unsatisfiability proof for kmin − 1 = 22 was only achieved later by CryptoMiniSAT =-=[8]-=- (cf. [4]). The corresponding value of kmin for the bottom matrix is currently unknown. In [2], Boyar and Peralta report that they have found linear straight-line programs of 30 lines for the bottom m... |

33 | Solving partial order constraints for LPO termination.
- Codish, Lagoon, et al.
- 2006
(Show Context)
Citation Context ... ∧ ψ(j, p) 1≤p<i We finally get an encoding δ in prenex normal form that can be used as input for a SAT solver (by dropping explicit existential quantification, encoding cardinality constraints using =-=[5, 1]-=-, and performing Tseitin’s transformation [14]). δ3(ℓ) = ∧ ⎛ ⎝fℓ,i → ∧ ⎞ (ψ(j, i) ↔ aℓ,j) ⎠ ∧ exactly1(fℓ,1, . . . , fℓ,k) 1≤i≤k 1≤j≤n δ = ∃b1,1. . . . ∃bk,n.∃c1,1. . . ....∃ck,k.∃f1,1. . . . ∃fm,k. β... |

28 | Testing positiveness of polynomials.
- Hong, Jakus
- 1998
(Show Context)
Citation Context .... . .,xn, our program should yield the correct result. Fortunately, we can now benefit from linearity of the operation ⊕ on GF(2), which means that the absolute positiveness criterion for polynomials =-=[10]-=- (a simple technique commonly used in automated termination provers, cf. e.g. [7]) is not only sound, but also complete. Essentially, the idea is that two linear forms compute the same function iff th... |

19 | A new combinational logic minimization technique with applications to cryptology. Experimental Algorithms
- Boyar, Peralta
- 2010
(Show Context)
Citation Context ...In Proc. SAT’10, LNCS, 2010. Supported by the G.I.F. grant 966-116.6 and the Danish Natural Science Research Council.While there are heuristic methods for finding short straight-line linear programs =-=[4]-=- (see also [3] for the corresponding patent application), to the best of our knowledge, there is no feasible method for finding an optimal solution. In this paper, we present an approach based on redu... |

11 | Cardinality Networks and Their Applications.
- Asın, Nieuwenhuis, et al.
- 2009
(Show Context)
Citation Context ... ∧ ψ(j, p) 1≤p<i We finally get an encoding δ in prenex normal form that can be used as input for a SAT solver (by dropping explicit existential quantification, encoding cardinality constraints using =-=[5, 1]-=-, and performing Tseitin’s transformation [14]). δ3(ℓ) = ∧ ⎛ ⎝fℓ,i → ∧ ⎞ (ψ(j, i) ↔ aℓ,j) ⎠ ∧ exactly1(fℓ,1, . . . , fℓ,k) 1≤i≤k 1≤j≤n δ = ∃b1,1. . . . ∃bk,n.∃c1,1. . . ....∃ck,k.∃f1,1. . . . ∃fm,k. β... |

11 |
Finding efficient circuits using SAT-solvers
- Kojevnikov, Kulikov, et al.
- 2009
(Show Context)
Citation Context ...n problem (“Is there a program of length k?”) to satisfiability of propositional logic. The reduction is performed in a way that every model found by the SAT solver represents a solution. Recent work =-=[11]-=- has shown that reductions to satisfiability problems are a promising approach for circuit synthesis. By restricting our attention to linear functions, we now obtain a polynomial-size encoding. The st... |

10 | Inferring network invariants automatically.
- Grinchtein, Leucker, et al.
- 2006
(Show Context)
Citation Context ...ust” 1500 variables. In this section we discuss three different approaches based on unary SAT encodings, on Pseudo-Boolean satisfiability, and on symmetry breaking. 6.1 Unary encodings As remarked by =-=[9]-=-, encoding arithmetic in unary representation instead of the more common binary (CPU-like) representation can be very beneficial for the performance of modern conflict-driven SAT solvers on the result... |

7 |
On the Shortest Linear Straight-Line Program for Computing Linear Forms
- Boyar, Matthews, et al.
- 2008
(Show Context)
Citation Context ...utes all yℓ. Note that here we are aiming at a (provably) optimal solution. This is opposed to allowing approximations with more lines than actually necessary, which is currently the state of the art =-=[2]-=-. As a step towards solving this optimization problem, first let us consider the corresponding decision problem: Given n variables x1, . . .,xn over GF(2), m linear forms yℓ = aℓ,1 · x1 ⊕ . . . ⊕ aℓ,n... |

1 |
A new technique for combinational circuit optimization and a new circuit for the S-Box for AES. Patent Application Number 61089998 filed with the U.S. Patent and Trademark Office
- Boyar, Peralta
- 2009
(Show Context)
Citation Context ...0, LNCS, 2010. Supported by the G.I.F. grant 966-116.6 and the Danish Natural Science Research Council.While there are heuristic methods for finding short straight-line linear programs [4] (see also =-=[3]-=- for the corresponding patent application), to the best of our knowledge, there is no feasible method for finding an optimal solution. In this paper, we present an approach based on reducing the assoc... |

1 | Optimizing the AES S-box using SAT - Fuhs, Schneider-Kamp |