# American Institute of Mathematical Sciences

doi: 10.3934/amc.2021067
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## Following Forrelation – quantum algorithms in exploring Boolean functions' spectra

 1 Applied Statistics Unit, Indian Statistical Institute, 203 B T Road, Kolkata 700 108, India 2 Indian Statistical Institute, 203 B T Road, Kolkata 700 108, India, University of Southern California, USA

*Corresponding author: Subhamoy Maitra

Received  October 2021 Early access January 2022

Here we revisit the quantum algorithms for obtaining Forrelation [Aaronson et al., 2015] values to evaluate some of the well-known cryptographically significant spectra of Boolean functions, namely the Walsh spectrum, the cross-correlation spectrum, and the autocorrelation spectrum. We introduce the existing 2-fold Forrelation formulation with bent duality-based promise problems as desirable instantiations. Next, we concentrate on the 3-fold version through two approaches. First, we judiciously set up some of the functions in 3-fold Forrelation so that given oracle access, one can sample from the Walsh Spectrum of $f$. Using this, we obtain improved results than what one can achieve by exploiting the Deutsch-Jozsa algorithm. In turn, it has implications in resiliency checking. Furthermore, we use a similar idea to obtain a technique in estimating the cross-correlation (and thus autocorrelation) value at any point, improving upon the existing algorithms. Finally, we tweak the quantum algorithm with the superposition of linear functions to obtain a cross-correlation sampling technique. This is the first cross-correlation sampling algorithm with constant query complexity to the best of our knowledge. This also provides a strategy to check if two functions are uncorrelated of degree $m$. We further modify this using Dicke states so that the time complexity reduces, particularly for constant values of $m$.

Citation: Suman Dutta, Subhamoy Maitra, Chandra Sekhar Mukherjee. Following Forrelation – quantum algorithms in exploring Boolean functions' spectra. Advances in Mathematics of Communications, doi: 10.3934/amc.2021067
##### References:
 [1] S. Aaronson, BQP and the polynomial hierarchy, In Proceedings of the Forty-Second ACM Symposium on Theory of Computing (STOC '10), Association for Computing Machinery, New York, NY, USA, (2010), 141–150, arXiv version, arXiv: 0910.4698, (2009). doi: 10.1145/1806689.1806711. [2] S. Aaronson and A. Ambainis, Forrelation: A problem that optimally separates quantum from classical computing, Siam J. Comput., 47 (2018), 982–1038. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC '15), Association for Computing Machinery, New York, NY, USA, DOI: https://doi.org/10.1145/2746539.2746547, (2015), 307–316, arXiv version, arXiv: 1411.5729, (2014). doi: 10.1137/15M1050902. [3] N. Bansal and M. Sinha, k-forrelation optimally separates quantum and classical query complexity, In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC '21), (2021), 1303–1316. arXiv version, arXiv: 2008.07003, (2020). doi: 10.1145/3406325.3451040. [4] D. Bera, S. Maitra and S. Tharrmashastha, Efficient quantum algorithms related to autocorrelation spectrum, In Progress in Cryptology-Indocrypt 2019, Lecture Notes in Computer Science, vol. 11898,415–432, Springer, (2019), arXiv version, arXiv: 1808.04448, (2019). doi: 10.1007/978-3-030-35423-7_21. [5] G. Brassard, P. Hoyer, M. Mosca and A. Tapp, Quantum amplitude amplification and estimation, Quantum Computation and Quantum Information, Samuel J. Lomonaco, Jr. (editor), AMS Contemporary Mathematics, vol. 305 (2002), 53–74, arXiv version, arXiv: quant-ph/0005055, (2000). doi: 10.1090/conm/305/05215. [6] C. Carlet, Two new classes of bent functions, Eurocrypt 1993, Lecture Notes in Computer Science, volume 765 (1994), Springer, 77–101. doi: 10.1007/3-540-48285-7_8. [7] K. Chakraborty and S. Maitra, Application of Grover's algorithm to check non-resiliency of a Boolean function, Cryptography and Communications, 8 (2016), 401-413.  doi: 10.1007/s12095-015-0156-3. [8] D. Deutsch and R. Jozsa, Rapid solution of problems by quantum computation, Proceedings of Royal Society London, 439 (1992), 553-558.  doi: 10.1098/rspa.1992.0167. [9] J. F. Dillon, Elementary Hadamard Difference Sets, Ph.D. thesis, U. of Maryland, 1974. Also see "Elementary Hadamard difference sets" in: Proc. Sixth Southeastern Conf. Combinatorics, Graph Theory, and Computing, Winnipeg, (1975). doi: 10.13016/M2MS3K194. [10] L. K. Grover, A fast quantum mechanical algorithm for database search, In Proceedings of the Twenty-Eighth ACM Symposium on Theory of Computing (STOC'96), Association for Computing Machinery, New York, (1996), 212–219, arXiv version, arXiv: quant-ph/9605043, (1996). doi: 10.1145/237814.237866. [11] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, North-Holland, Amsterdam, 1977. [12] C. S. Mukherjee, S. Maitra, V. Gaurav and D. Roy, On actual preparing Dicke states on a quantum computer, In IEEE Transactions on Quantum Engineering, 1 (2020), Art no. 3102517, 1-17, arXiv version, arXiv: 2007.01681, (2020). doi: 10.1109/TQE.2020.3041479. [13] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Edition, Cambridge University Press, 2011. [14] M. Roetteler, Quantum algorithms for highly non-linear Boolean functions, In the Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), (2010), 448–457, arXiv version, arXiv: 0811.3208, (2009). doi: 10.1137/1.9781611973075.37. [15] P. Sarkar and S. Maitra, Cross-correlation analysis of cryptographically useful Boolean functions and S-boxes, Theory of Computing Systems, 35 (2002), 39-57.  doi: 10.1007/s00224-001-1019-1. [16] P. Sarkar and S. Maitra, Construction of nonlinear resilient Boolean functions using 'small' affine functions, IEEE Transactions on Information Theory, 50 (2004), 2185-2193.  doi: 10.1109/TIT.2004.833366. [17] D. R. Simon, On the power of quantum computation, SIAM Journal on Computing, 26 (1997), 1474-1483.  doi: 10.1137/S00975397962986371. [18] P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Journal on Computing, 26 (1997), 1484–1509, arXiv version, arXiv: quant-ph/9508027, (1996). doi: 10.1137/S0097539795293172. [19] P. Stanica, B. Mandal and S. Maitra, The connection between quadratic bent-negabent functions and the Kerdock code, Appl. Algebra Eng. Commun. Comput., 30 (2019), 387-401.  doi: 10.1007/s00200-019-00380-4. [20] A. Tal, Towards optimal separations between quantum and randomized query complexities, IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), Durham, NC, USA, (2020), 228-239, arXiv version, arXiv: 1912.12561, (2019). doi: 10.1109/FOCS46700.2020.00030. [21] G.-Z. Xiao and J. L. Massey, A spectral characterization of correlation immune combining functions, IEEE Transactions on Information Theory, 34 (1988), 569–571. doi: 10.1109/18.6037.

show all references

##### References:
 [1] S. Aaronson, BQP and the polynomial hierarchy, In Proceedings of the Forty-Second ACM Symposium on Theory of Computing (STOC '10), Association for Computing Machinery, New York, NY, USA, (2010), 141–150, arXiv version, arXiv: 0910.4698, (2009). doi: 10.1145/1806689.1806711. [2] S. Aaronson and A. Ambainis, Forrelation: A problem that optimally separates quantum from classical computing, Siam J. Comput., 47 (2018), 982–1038. In Proceedings of the 47th Annual ACM Symposium on Theory of Computing (STOC '15), Association for Computing Machinery, New York, NY, USA, DOI: https://doi.org/10.1145/2746539.2746547, (2015), 307–316, arXiv version, arXiv: 1411.5729, (2014). doi: 10.1137/15M1050902. [3] N. Bansal and M. Sinha, k-forrelation optimally separates quantum and classical query complexity, In Proceedings of the 53rd Annual ACM SIGACT Symposium on Theory of Computing (STOC '21), (2021), 1303–1316. arXiv version, arXiv: 2008.07003, (2020). doi: 10.1145/3406325.3451040. [4] D. Bera, S. Maitra and S. Tharrmashastha, Efficient quantum algorithms related to autocorrelation spectrum, In Progress in Cryptology-Indocrypt 2019, Lecture Notes in Computer Science, vol. 11898,415–432, Springer, (2019), arXiv version, arXiv: 1808.04448, (2019). doi: 10.1007/978-3-030-35423-7_21. [5] G. Brassard, P. Hoyer, M. Mosca and A. Tapp, Quantum amplitude amplification and estimation, Quantum Computation and Quantum Information, Samuel J. Lomonaco, Jr. (editor), AMS Contemporary Mathematics, vol. 305 (2002), 53–74, arXiv version, arXiv: quant-ph/0005055, (2000). doi: 10.1090/conm/305/05215. [6] C. Carlet, Two new classes of bent functions, Eurocrypt 1993, Lecture Notes in Computer Science, volume 765 (1994), Springer, 77–101. doi: 10.1007/3-540-48285-7_8. [7] K. Chakraborty and S. Maitra, Application of Grover's algorithm to check non-resiliency of a Boolean function, Cryptography and Communications, 8 (2016), 401-413.  doi: 10.1007/s12095-015-0156-3. [8] D. Deutsch and R. Jozsa, Rapid solution of problems by quantum computation, Proceedings of Royal Society London, 439 (1992), 553-558.  doi: 10.1098/rspa.1992.0167. [9] J. F. Dillon, Elementary Hadamard Difference Sets, Ph.D. thesis, U. of Maryland, 1974. Also see "Elementary Hadamard difference sets" in: Proc. Sixth Southeastern Conf. Combinatorics, Graph Theory, and Computing, Winnipeg, (1975). doi: 10.13016/M2MS3K194. [10] L. K. Grover, A fast quantum mechanical algorithm for database search, In Proceedings of the Twenty-Eighth ACM Symposium on Theory of Computing (STOC'96), Association for Computing Machinery, New York, (1996), 212–219, arXiv version, arXiv: quant-ph/9605043, (1996). doi: 10.1145/237814.237866. [11] F. J. MacWilliams and N. J. A. Sloane, The Theory of Error Correcting Codes, North-Holland, Amsterdam, 1977. [12] C. S. Mukherjee, S. Maitra, V. Gaurav and D. Roy, On actual preparing Dicke states on a quantum computer, In IEEE Transactions on Quantum Engineering, 1 (2020), Art no. 3102517, 1-17, arXiv version, arXiv: 2007.01681, (2020). doi: 10.1109/TQE.2020.3041479. [13] M. A. Nielsen and I. L. Chuang, Quantum Computation and Quantum Information, 10th Anniversary Edition, Cambridge University Press, 2011. [14] M. Roetteler, Quantum algorithms for highly non-linear Boolean functions, In the Proceedings of the 21st Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), (2010), 448–457, arXiv version, arXiv: 0811.3208, (2009). doi: 10.1137/1.9781611973075.37. [15] P. Sarkar and S. Maitra, Cross-correlation analysis of cryptographically useful Boolean functions and S-boxes, Theory of Computing Systems, 35 (2002), 39-57.  doi: 10.1007/s00224-001-1019-1. [16] P. Sarkar and S. Maitra, Construction of nonlinear resilient Boolean functions using 'small' affine functions, IEEE Transactions on Information Theory, 50 (2004), 2185-2193.  doi: 10.1109/TIT.2004.833366. [17] D. R. Simon, On the power of quantum computation, SIAM Journal on Computing, 26 (1997), 1474-1483.  doi: 10.1137/S00975397962986371. [18] P. W. Shor, Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer, SIAM Journal on Computing, 26 (1997), 1484–1509, arXiv version, arXiv: quant-ph/9508027, (1996). doi: 10.1137/S0097539795293172. [19] P. Stanica, B. Mandal and S. Maitra, The connection between quadratic bent-negabent functions and the Kerdock code, Appl. Algebra Eng. Commun. Comput., 30 (2019), 387-401.  doi: 10.1007/s00200-019-00380-4. [20] A. Tal, Towards optimal separations between quantum and randomized query complexities, IEEE 61st Annual Symposium on Foundations of Computer Science (FOCS), Durham, NC, USA, (2020), 228-239, arXiv version, arXiv: 1912.12561, (2019). doi: 10.1109/FOCS46700.2020.00030. [21] G.-Z. Xiao and J. L. Massey, A spectral characterization of correlation immune combining functions, IEEE Transactions on Information Theory, 34 (1988), 569–571. doi: 10.1109/18.6037.
Quantum circuit for implementing the $2$-fold Forrelation problem using $2$ queries
Quantum circuit for implementing the $3$-fold Forrelation problem using $3$ sequential queries
Quantum circuit for implementing the $3$-fold Forrelation problem using $2$ parallel queries
Sampling probabilities of Walsh transform using different algorithms
Quantum circuit for implementing Algorithm 1
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