Abstract
\(\textit{HN}\)-transforms, which have been proposed as generalizations of Hadamard transforms, are constructed by tensoring Hadamard and nega-Hadamard kernels in any order. We show that all the \(2^n\) possible \(\textit{HN}\)-spectra of a Boolean function in n variables, each containing \(2^n\) elements (i.e., in total \(2^{2n}\) values in transformed domain) can be computed in \(O(2^{2n})\) time (more specific with little less than \(2^{2n+1}\) arithmetic operations). We propose a generalization of Deutsch-Jozsa algorithm, by employing \(\textit{HN}\)-transforms, which can be used to distinguish different classes of Boolean functions over and above what is possible by the traditional Deutsch-Jozsa algorithm.
S. Maitra is supported by the project “Cryptography & Cryptanalysis: How far can we bridge the gap between Classical and Quantum Paradigm”, awarded by the Scientific Research Council of the Department of Atomic Energy (DAE-SRC), the Board of Research in Nuclear Sciences (BRNS).
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References
Aumasson, J.-P., Dinur, I., Meier, W., Shamir, A.: Cube testers and key recovery attacks on reduced-round MD6 and trivium. In: Dunkelman, O. (ed.) FSE 2009. LNCS, vol. 5665, pp. 1–22. Springer, Heidelberg (2009). doi:10.1007/978-3-642-03317-9_1
Cusick, T.W., Stănică, P.: Cryptographic Boolean Functions and Applications, 2nd edn. Academic Press, San Diego (2017). 1st edn. (2009)
Danielsen, L.E., Parker, M.G.: Spectral orbits and peak-to-average power ratio of boolean functions with respect to the \(\mathit{I},\mathit{H},\mathit{N}^{\mathit{n}}\) transform. In: Helleseth, T., Sarwate, D., Song, H.-Y., Yang, K. (eds.) SETA 2004. LNCS, vol. 3486, pp. 373–388. Springer, Heidelberg (2005). doi:10.1007/11423461_28
Danielsen, L.E.: On connections between graphs, codes, quantum states, and Boolean functions. Ph.D. thesis, Department of Informatics, The Selmer Center, University of Bergen, Norway (2008)
Deutsch, D., Jozsa, R.: Rapid solution of problems by quantum computation. Proc. Roy. Soc. Lond. A439, 553–558 (1992)
Dillon, J.F.: Elementary Hadamard difference sets. Ph.D. thesis, University of Maryland (1974)
Dinur, I., Shamir, A.: Cube attacks on tweakable black box polynomials. In: Joux, A. (ed.) EUROCRYPT 2009. LNCS, vol. 5479, pp. 278–299. Springer, Heidelberg (2009). doi:10.1007/978-3-642-01001-9_16. See also: Cube Attacks on Tweakable Black Box Polynomials. http://eprint.iacr.org/2008/385.pdf
Gangopadhyay, S., Pasalic, E., Stănică, P.: A note on generalized bent criteria for boolean functions. IEEE Trans. Inf. Theor. 59(5), 3233–3236 (2013)
Gangopadhyay, S., Gangopadhyay, A.K., Pollatos, S., Stănică, P.: Cryptographic Boolean functions with biased inputs. Crypt. Commun. Discrete Struct. Seq. 9, 301–314 (2017). doi:10.1007/s12095-015-0174-1
Helleseth, T., Kløve, T., Mvkkeltveit, J.: On the covering radius of binary codes. IEEE Trans. Inf. Theor. 24(5), 627–628 (1978)
Knudsen, L.R.: Truncated and higher order differentials. In: Preneel, B. (ed.) FSE 1994. LNCS, vol. 1008, pp. 196–211. Springer, Heidelberg (1995). doi:10.1007/3-540-60590-8_16
Litsyn, S., Shpunt, A.: On the distribution of Boolean function nonlinearity. SIAM J. Discrete Math. 23(1), 79–95 (2008)
Maitra, S., Mukhopadhyay, P.: Deutsch-Jozsa algorithm revisited in the domain of cryptographically significant boolean functions. Int. J. Quantum Inf. 3(2), 359–370 (2005)
Matsui, M.: Linear cryptanalysis method for DES cipher. In: Helleseth, T. (ed.) EUROCRYPT 1993. LNCS, vol. 765, pp. 386–397. Springer, Heidelberg (1994). doi:10.1007/3-540-48285-7_33
Meier, W., Staffelbach, O.: Fast correlation attacks on stream ciphers. In: Barstow, D., Brauer, W., Brinch Hansen, P., Gries, D., Luckham, D., Moler, C., Pnueli, A., Seegmüller, G., Stoer, J., Wirth, N., Günther, C.G. (eds.) EUROCRYPT 1988. LNCS, vol. 330, pp. 301–314. Springer, Heidelberg (1988). doi:10.1007/3-540-45961-8_28
Parker, M.G.: Generalised \(S\)-box nonlinearity. NESSIE Public Document, 11.02.03: NES/DOC/UIB/WP5/020/A
Parker, M.G., Pott, A.: On boolean functions which are bent and negabent. In: Golomb, S.W., Gong, G., Helleseth, T., Song, H.-Y. (eds.) SSC 2007. LNCS, vol. 4893, pp. 9–23. Springer, Heidelberg (2007). doi:10.1007/978-3-540-77404-4_2
Patterson, N.J., Wiedemann, D.H.: The covering radius of the \((2^{15}, 16)\) Reed-Muller code is at least 16276. IEEE Trans. Inf. Theor. 29(3), 354–356 (1983). See also correction: IEEE Trans. Inf. Theor. 36(2), 443 (1990)
Riera, C.: Spectral properties of Boolean functions, graphs and graph states. Ph.D. thesis, University of Bergen (2005)
Riera, C., Parker, M.G.: Generalized bent criteria for Boolean functions. IEEE Trans. Inf. Theor. 52(9), 4142–4159 (2006)
Schmidt, K.-U., Parker, M.G., Pott, A.: Negabent functions in the Maiorana–McFarland class. In: Golomb, S.W., Parker, M.G., Pott, A., Winterhof, A. (eds.) SETA 2008. LNCS, vol. 5203, pp. 390–402. Springer, Heidelberg (2008). doi:10.1007/978-3-540-85912-3_34
Siegenthaler, T.: Decrypting a class of stream ciphers using ciphertext only. IEEE Trans. Comput. 34(1), 81–85 (1985)
Stănică, P., Maitra, S.: Rotation symmetric Boolean functions - count and cryptographic properties. Disc. Appl. Math. 156, 1567–1580 (2008)
Stănică, P., Gangopadhyay, S., Chaturvedi, A., Kar-Gangopadhyay, A., Maitra, S.: Investigations on bent and negabent functions via the nega-Hadamard transform. IEEE Trans. Inf. Theor. 58(6), 4065–4072 (2012)
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Gangopadhyay, S., Maitra, S., Sinha, N., Stănică, P. (2017). Quantum Algorithms Related to \(\textit{HN}\)-Transforms of Boolean Functions. In: El Hajji, S., Nitaj, A., Souidi, E. (eds) Codes, Cryptology and Information Security. C2SI 2017. Lecture Notes in Computer Science(), vol 10194. Springer, Cham. https://doi.org/10.1007/978-3-319-55589-8_21
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