Abstract
We show that degree-d block-symmetric polynomials in n variables modulo any odd p correlate with parity exponentially better than degree-d symmetric polynomials, if \({n \ge cd^2 {\rm log} d}\) and \({d \in [0.995 \cdot p^t - 1,p^t)}\) for some \({t \ge 1}\) and some \({c > 0}\) that depends only on p. For these infinitely many degrees, our result solves an open problem raised by a number of researchers including Alon & Beigel (IEEE conference on computational complexity (CCC), pp 184–187, 2001). The only previous case for which this was known was d = 2 and p = 3 (Green in J Comput Syst Sci 69(1):28–44, 2004).
The result is obtained through the development of a theory we call spectral analysis of symmetric correlation, which originated in works of Cai et al. (Math Syst Theory 29(3):245–258, 1996) and Green (Theory Comput Syst 32(4):453–466, 1999). In particular, our result follows from a detailed analysis of the correlation of symmetric polynomials, which is determined up to an exponentially small relative error when \({d = p^t-1}\).
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References
Scott Aaronson & Avi Wigderson (2008). Algebrization: a new barrier in complexity theory. In 40th ACM Symp. on the Theory of Computing (STOC), 731–740.
Noga Alon & Richard Beigel (2001). Lower bounds for approximations by low degree polynomials over Z m . In IEEE Conf. on Computational Complexity (CCC), 184–187.
Kazuyuki Amano (2010) Researching the Complexity of Boolean Functions with Computers. Bulletin of the EATCS 101: 64–91
László Babai, Noam Nisan & Márió Szegedy (1992). Multiparty protocols, pseudorandom generators for logspace, and time-space trade-offs. J. of Computer and System Sciences 45(2), 204–232. ISSN 0022-0000.
Theodore Baker, John Gill & Robert Solovay (1975). Relativizations of the P=?NP question. SIAM J. Comput. 4(4), 431–442. ISSN 1095-7111.
Nayantara Bhatnagar, Parikshit Gopalan, Richard J. Lipton (2006) -symmetric polynomials over Z m and simultaneous communication protocols. J. of Computer and System Sciences 72(2): 252–285
Joppe Bos & Marcelo Kaihara (2010). PlayStation 3 computing breaks 260 barrier: 112-bit prime ECDLP solved.
Jean Bourgain (2005). Estimation of certain exponential sums arising in complexity theory. C. R. Math. Acad. Sci. Paris 340(9), 627–631. ISSN 1631-073X.
Jin-Yi Cai, Frederic Green, Thomas Thierauf (1996) On the Correlation of Symmetric Functions. Mathematical Systems Theory 29(3): 245–258
Eduardo Dueñez, Steven J. Miller, Amitabha Roy & Howard Straubing (2006). Incomplete quadratic exponential sums in several variables. J. Number Theory 116(1), 168–199. ISSN 0022-314X.
Frederic Green (1999) Exponential Sums and Circuits with a Single Threshold Gate and Mod-Gates. Theory Comput. Syst. 32(4): 453–466
Frederic Green (2004) The correlation between parity and quadratic polynomials mod 3. J. of Computer and System Sciences 69(1): 28–44
Frederic Green, Amitabha Roy (2010) Uniqueness of Optimal Mod 3 Circuits for Parity. Journal of Number Theory 130: 961–975
Frederic Green, Amitabha Roy & Howard Straubing (2005). Bounds on an exponential sum arising in Boolean circuit complexity. C. R. Math. Acad. Sci. Paris 341(5), 279–282. ISSN 1631-073X.
Stanislaw Radziszowski (2014). Small Ramsey Numbers. The Electronic Journal of Combinatorics Dynamic Surveys, DS1.14, 1–94.
Alexander Razborov (1987). Lower bounds on the dimension of schemes of bounded depth in a complete basis containing the logical addition function. Mat. Zametki 41(4), 598–607. English translation in Mathematical Notes of the Academy of Sci. of the USSR, 41(4):333-338, 1987.
Alexander Razborov, Steven Rudich (1997) Natural Proofs. J. of Computer and System Sciences 55(1): 24–35
Roman Smolensky (1987). Algebraic methods in the theory of lower bounds for Boolean circuit complexity. In 19th ACM Symp. on the Theory of Computing (STOC), 77–82. ACM.
Roman Smolensky (1993). On Representations by Low-Degree Polynomials. In 34th IEEE IEEE Symp. on Foundations of Computer Science (FOCS), 130–138.
Emanuele Viola (2009) On the power of small-depth computation. Foundations and Trends in Theoretical Computer Science 5(1): 1–72
Emanuele Viola (2013). Challenges in computational lower bounds. Available at http://www.ccs.neu.edu/home/viola/.
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Green, F., Kreymer, D. & Viola, E. Block-symmetric polynomials correlate with parity better than symmetric. comput. complex. 26, 323–364 (2017). https://doi.org/10.1007/s00037-017-0153-3
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DOI: https://doi.org/10.1007/s00037-017-0153-3