Solving Multivariate Polynomial Systems
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1 Solving Multivariate Polynomial Systems Presented by: Bo-Yin Yang work with Lab of Yang and Cheng, and Charles Bouillaguet, ENS Institute of Information Science and TWISC, Academia Sinica Taipei, Taiwan Feb. 9, 2010
2 Most Practical Generic Solution for F 2? How to solve a generic system of m equations, n variables over F 2, not too overdetermined, n mid-sized, say 50 or 70. B.-Y. Yang (IIS-TWISC@Sinica) Solving MQ Feb. 9, / 6
3 Most Practical Generic Solution for F 2? How to solve a generic system of m equations, n variables over F 2, not too overdetermined, n mid-sized, say 50 or 70. MQ (Multivariate Quadratics) Problem MQ(q; n, m) Want a solution for n variables in m quadratic equations, All coefficients and variables in F q, if one exists. NP-Hard even when q = 2 B.-Y. Yang (IIS-TWISC@Sinica) Solving MQ Feb. 9, / 6
4 Most Practical Generic Solution for F 2? How to solve a generic system of m equations, n variables over F 2, not too overdetermined, n mid-sized, say 50 or 70. MQ (Multivariate Quadratics) Problem MQ(q; n, m) Want a solution for n variables in m quadratic equations, All coefficients and variables in F q, if one exists. NP-Hard even when q = 2 Conjectured Probabilistically Hard [Berbain-Gilbert-Patarin] B.-Y. Yang (IIS-TWISC@Sinica) Solving MQ Feb. 9, / 6
5 Most Practical Generic Solution for F 2? How to solve a generic system of m equations, n variables over F 2, not too overdetermined, n mid-sized, say 50 or 70. MQ (Multivariate Quadratics) Problem MQ(q; n, m) Want a solution for n variables in m quadratic equations, All coefficients and variables in F q, if one exists. NP-Hard even when q = 2 Conjectured Probabilistically Hard [Berbain-Gilbert-Patarin] There has been many recent advances in system-solving, what is the best practical way to solve this? F 4? F 5? XL variant? B.-Y. Yang (IIS-TWISC@Sinica) Solving MQ Feb. 9, / 6
6 Solving for 32 F 2 Variables from Quadratics MAGMA-2.15 results More equations Easier system: 32 Equations: 55 Gigabytes, 2.2 days on 2.2GHz Opteron core 64 Equations: 2.5 Gigabytes, 3 hours on 2.2GHz Opteron core 320 Equations: 0.2 Gigabytes, 4.1 seconds on a 2.2GHz Opteron core Can we do better? B.-Y. Yang (IIS-TWISC@Sinica) Solving MQ Feb. 9, / 6
7 Solving for 32 F 2 Variables from Quadratics MAGMA-2.15 results More equations Easier system: 32 Equations: 55 Gigabytes, 2.2 days on 2.2GHz Opteron core 64 Equations: 2.5 Gigabytes, 3 hours on 2.2GHz Opteron core 320 Equations: 0.2 Gigabytes, 4.1 seconds on a 2.2GHz Opteron core Can we do better? A Smarter Brute-Force We implemented an idea brought to us by Charles Bouillaguet 2.2 GHz Opteron core: seconds nvidia GTX 280 (1.296 GHz): 0.05 seconds B.-Y. Yang (IIS-TWISC@Sinica) Solving MQ Feb. 9, / 6
8 Larger/Higher Systems? MAGMA-2.15 performance Solves 20 cubics in 20 F 2 variables in about 9000 s on Opteron 2.2GHz; runs out of memory with 23 variables and 23 equations. B.-Y. Yang (IIS-TWISC@Sinica) Solving MQ Feb. 9, / 6
9 Larger/Higher Systems? MAGMA-2.15 performance Solves 20 cubics in 20 F 2 variables in about 9000 s on Opteron 2.2GHz; runs out of memory with 23 variables and 23 equations. 36 F 2 vars in 36+ eqs, 1 core, C2Q GHz Quadratics: 9.0 s Cubics: 23.2 s Quartics: 55.8 s B.-Y. Yang (IIS-TWISC@Sinica) Solving MQ Feb. 9, / 6
10 Larger/Higher Systems? MAGMA-2.15 performance Solves 20 cubics in 20 F 2 variables in about 9000 s on Opteron 2.2GHz; runs out of memory with 23 variables and 23 equations. 36 F 2 vars in 36+ eqs, 1 core, C2Q GHz Quadratics: 9.0 s Cubics: 23.2 s Quartics: 55.8 s 48 F 2 vars in 48+ eqs, GTX 280 Quadratics: 41 mins Cubics: 73 mins Quartics: 280 mins B.-Y. Yang (IIS-TWISC@Sinica) Solving MQ Feb. 9, / 6
11 Why? o(1) 0 F 4 solving n F 2 equations in as many variables should take 2 (0.78+o(1))n time if we can apply sparse matrix techniques. But won t beat brute-force in number of logical ops until n = 200. Memory Effects Systems with Large Memory are slower: some claims O(M 1/2 ) slowdown. Better Enumeration thru Gray Code Via tracking all successive differentials, solving for n F 2 variables from degree-d systems takes O(2 n d polylog(n)) time. B.-Y. Yang (IIS-TWISC@Sinica) Solving MQ Feb. 9, / 6
12 What Now? Thanks to the Other People Charles Bouillaguet of ENS, Hsieh-Chung Kevin Chen, Ming-Shing Chen, Chen-Mou Cheng, Tony Chou, Ruben Niederhagen at our Lab. Take-Away Point For not-too-overdetermined F 2 systems in the practical range, Brute Force works better than F 4 and other Gröbner basis solvers. This affects, for example, security guarantees of QUAD stream ciphers. Future Results will be submitted and on eprint archive soon. B.-Y. Yang (IIS-TWISC@Sinica) Solving MQ Feb. 9, / 6
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