Juan Zhao
ABSTRACT
Solving Boolean polynomial systems as an important aspect of symbolic computation, plays a fundamental role in various real applications. Although there exist many efficient sequential algorithms for solving Boolean polynomial systems, they are inefficient or even unavailable when the problem scale becomes large, due to the computational complexity of the problem and the limited processing capability of a single node. In this paper we propose an efficient parallel characteristic set method called PBCS for solving Boolean polynomial systems under the high-performance computing environment. Specifically, PBCS takes full advantage of the state-of-the-art characteristic set method and achieves load balancing by dynamically reallocating tasks. Moreover, the performance is further improved by optimizing the parameter setting. Extensive experiments are conducted to demonstrate that PBCS is efficient and scalable for solving Boolean equations, especially for the equations rasing from stream ciphers that have block triangular structure. In addition, the algorithm has good scalability and can be extended to the size of thousands CPU cores with a stable speedup.
- Magali Bardet, Jean-Charles Faugere, and Bruno Salvy. 2004. On the complexity of Gröbner basis computation of semi-regular overdetermined algebraic equations. In Proceedings of the International Conference on Polynomial System Solving. 71--74.Google Scholar
- Andrey Bogdanov, Lars R Knudsen, Gregor Leander, Christof Paar, Axel Poschmann, Matthew JB Robshaw, Yannick Seurin, and Charlotte Vikkelsoe. 2007. PRESENT: An ultra-lightweight block cipher. In International Workshop on Cryptographic Hardware and Embedded Systems. Springer, 450--466. Google ScholarDigital Library
- Driss Bouziane, Abdelilah Kandri Rody, and Hamid Maârouf. 2001. Unmixed-dimensional decomposition of a finitely generated perfect differential ideal. Journal of Symbolic Computation 31, 6 (2001), 631--649. Google ScholarDigital Library
- Michael Brickenstein and Alexander Dreyer. 2009. PolyBoRi: A framework for Gröbner-basis computations with Boolean polynomials. Journal of Symbolic Computation 44, 9 (2009), 1326--1345. Google ScholarDigital Library
- S Cook. 2004. From Satisfiability to Proof Complexity and Bounded Arithmetic. SAT (2004).Google Scholar
- Stephen Cook and Phuong Nguyen. 2010. Logical foundations of proof complexity. Cambridge University Press. Google ScholarDigital Library
- Nicolas Courtois, Alexander Klimov, Jacques Patarin, and Adi Shamir. 2000. Efficient algorithms for solving overdefined systems of multivariate polynomial equations. In International Conference on the Theory and Applications of Cryptographic Techniques. Springer, 392--407. Google ScholarDigital Library
- Xavier Dahan, Marc Moreno Maza, Eric Schost, Wenyuan Wu, and Yuzhen Xie. 2005. Lifting techniques for triangular decompositions. In Proceedings of the 2005 international symposium on Symbolic and algebraic computation. ACM, 108--115. Google ScholarDigital Library
- Christophe De Canniere. 2006. Trivium: A stream cipher construction inspired by block cipher design principles. In International Conference on Information Security. Springer, 171--186. Google ScholarDigital Library
- Christian Eder and Bjarke Hammersholt Roune. 2013. Signature rewriting in gröbner basis computation. In Proceedings of the 38th International Symposium on Symbolic and Algebraic Computation. ACM, 331--338. Google ScholarDigital Library
- Jean-Charles Faugere. 1999. A new efficient algorithm for computing Gröbner bases (F4). Journal of pure and applied algebra 139, 1--3 (1999), 61--88.Google ScholarCross Ref
- Jean-Charles Faugere and Gwénolé Ars. 2003. An algebraic cryptanalysis of nonlinear filter generators using Gröbner bases. Ph.D. Dissertation. INRIA.Google Scholar
- Jean-Charles Faugère and Sylvain Lachartre. 2010. Parallel Gaussian Elimination for Gröbner bases computations in finite fields. In Proceedings of the 4th International Workshop on Parallel and Symbolic Computation. ACM, 89--97. Google ScholarDigital Library
- Shuhong Gao, Frank Volny IV, and Mingsheng Wang. 2016. A new framework for computing Gröbner bases. Mathematics of computation 85, 297 (2016), 449--465.Google Scholar
- Xiao-Shan Gao and Zhenyu Huang. 2012. Characteristic set algorithms for equation solving in finite fields. Journal of Symbolic Computation 47, 6 (2012), 655--679. Google ScholarDigital Library
- Xiao-Shan Gao, Joris Van Der Hoeven, Chun-Ming Yuan, and Gui-Lin Zhang. 2009. Characteristic set method for differential--difference polynomial systems. Journal of Symbolic Computation 44, 9 (2009), 1137--1163. Google ScholarDigital Library
- Zhenyu Huang and Dongdai Lin. 2011. Attacking Bivium and Trivium with the characteristic set method. In International Conference on Cryptology in Africa. Springer, 77--91. Google ScholarDigital Library
- Zhenyu Huang and Dongdai Lin. 2017. Solving polynomial systems with noise over F2: Revisited. Theoretical Computer Science 676 (2017), 52--68. Google ScholarDigital Library
- Zhenyu Huang, Yao Sun, and Dongdai Lin. 2014. On the efficiency of solving boolean polynomial systems with the characteristic set method. arXiv preprint arXiv:1405.4596 (2014).Google Scholar
- Riccardo Murri. 2011. A novel parallel algorithm for Gaussian Elimination of sparse unsymmetric matrices. In International Conference on Parallel Processing and Applied Mathematics. Springer, 183--193. Google ScholarDigital Library
- Jorge Nakahara, Pouyan Sepehrdad, Bingsheng Zhang, and Meiqin Wang. 2009. Linear (hull) and algebraic cryptanalysis of the block cipher PRESENT. In International Conference on Cryptology and Network Security. Springer, 58--75. Google ScholarDigital Library
- Havard Raddum. 2006. Cryptanalytic results on Trivium. eSTREAM, ECRYPT Stream Cipher Project, Report 39 (2006), 2006.Google Scholar
- Bjarke Hammersholt Roune and Michael Stillman. 2012. Practical Gröbner basis computation. In Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation. ACM, 203--210.Google ScholarDigital Library
- Fabio Somenzi. 2015. CUDD: CU decision diagram package release 3.0. 0. University of Colorado at Boulder (2015).Google Scholar
- Yong-Wei Wu, Guang-Wen Yang, Hong Yang, Wei-Min Zheng, and Dong-Dai Lin. 2005. Distributed computing model for Wu's method. Ruan Jian Xue Bao (J. Softw.) 16, 3 (2005), 384--391.Google Scholar
Index Terms
- PBCS: An Efficient Parallel Characteristic Set Method for Solving Boolean Polynomial Systems
Recommendations
Radiation modeling using the Uintah heterogeneous CPU/GPU runtime system
XSEDE '12: Proceedings of the 1st Conference of the Extreme Science and Engineering Discovery Environment: Bridging from the eXtreme to the campus and beyondThe Uintah Computational Framework was developed to provide an environment for solving fluid-structure interaction problems on structured adaptive grids on large-scale, long-running, data-intensive problems. Uintah uses a combination of fluid-flow ...
A cluster for CS education in the manycore era
SIGCSE '11: Proceedings of the 42nd ACM technical symposium on Computer science educationTraditional Beowulf clusters have been homogeneous platforms for distributed-memory MIMD parallelism. However, the shift to multicore architectures has made shared-memory MIMD parallelism increasingly important, and inexpensive manycore GPGPUs have ...
Model checking with generalized Rabin and Fin-less automata
In the automata theoretic approach to explicit state LTL model checking, the synchronized product of the model and an automaton that represents the negated formula is checked for emptiness. In practice, a (transition-based generalized) B chi automaton (...
Comments