Efficient Quantum Algorithms Related to Autocorrelation Spectrum

In this paper, we propose efficient probabilistic algorithms for several problems regarding the autocorrelation spectrum. First, we present a quantum algorithm that samples from the Walsh spectrum of any derivative of f(). Informally, the autocorrelation coefficient of a Boolean function f() at some point a measures the average correlation among the values f(x) and \(f(x \oplus a)\). The derivative of a Boolean function is an extension of autocorrelation to correlation among multiple values of f(). The Walsh spectrum is well-studied primarily due to its connection to the quantum circuit for the Deutsch-Jozsa problem. We extend the idea to “Higher-order Deutsch-Jozsa” quantum algorithm to obtain points corresponding to large absolute values in the Walsh spectrum of a certain derivative of f(). Further, we design an algorithm to sample the input points according to squares of the autocorrelation coefficients. Finally we provide a different set of algorithms for estimating the square of a particular coefficient or cumulative sum of their squares.

[1]  J. Stuart.  PROCEEDINGS - PART II , 1993 .

[2]  Lov K. Grover.  A fast quantum mechanical algorithm for database search , 1996, STOC '96.

[3]  Yishay Mansour,et al.  Learning Boolean Functions via the Fourier Transform , 1994 .

[4]  Theodore J. Yoder,et al.  Fixed-point quantum search with an optimal number of queries. , 2014, Physical review letters.

[5]  Subhamoy Maitra,et al.  Deutsch-Jozsa Algorithm Revisited in the Domain of Cryptographically Significant Boolean Functions , 2004 .

[6]  G. Brassard,et al.  Quantum Amplitude Amplification and Estimation , 2000, quant-ph/0005055.

[7]  Martin Hell,et al.  Grain: a stream cipher for constrained environments , 2007, Int. J. Wirel. Mob. Comput..

[8]  Claude E. Shannon,et al.  The mathematical theory of communication , 1950 .

[9]  Daowen Qiu,et al.  Quantum algorithms on Walsh transform and Hamming distance for Boolean functions , 2018, Quantum Inf. Process..

[10]  María Naya-Plasencia,et al.  An Efficient Quantum Collision Search Algorithm and Implications on Symmetric Cryptography , 2017, ASIACRYPT.

[11]  María Naya-Plasencia,et al.  Breaking Symmetric Cryptosystems Using Quantum Period Finding , 2016, CRYPTO.

[12]  D. Deutsch,et al.  Rapid solution of problems by quantum computation , 1992, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences.

[13]  Martin Rötteler,et al.  Post-Quantum Cryptography , 2015, Lecture Notes in Computer Science.

[14]  Debajyoti Bera,et al.  Error reduction of quantum algorithms , 2019, ArXiv.

[15]  Lov K. Grover.  Quantum Computers Can Search Rapidly by Using Almost Any Transformation , 1998 .

[16]  Gregor Leander,et al.  Grover Meets Simon - Quantumly Attacking the FX-construction , 2017, ASIACRYPT.

[17]  Oded Goldreich.  Foundations of Cryptography: Volume 1 , 2006 .

[18]  María Naya-Plasencia,et al.  Quantum Differential and Linear Cryptanalysis , 2015, IACR Trans. Symmetric Cryptol..

[19]  Xuejia Lai.  Higher Order Derivatives and Differential Cryptanalysis , 1994 .

[20]  Raymond Laflamme,et al.  An Introduction to Quantum Computing , 2007, Quantum Inf. Comput..

[21]  Gilles Brassard,et al.  An exact quantum polynomial-time algorithm for Simon's problem , 1997, Proceedings of the Fifth Israeli Symposium on Theory of Computing and Systems.

[22]  Deng Tang,et al.  Construction of $n$ -Variable ( $n\equiv 2 \bmod 4$ ) Balanced Boolean Functions With Maximum Absolute Value in Autocorrelation Spectra $< 2^{\frac {n}2}$ , 2018, IEEE Transactions on Information Theory.

[23]  Serge Vaudenay,et al.  Links Between Differential and Linear Cryptanalysis , 1994, EUROCRYPT.

[24]  Ryan O'Donnell,et al.  Analysis of Boolean Functions , 2014, ArXiv.

[25]  Santanu Sarkar,et al.  Observing biases in the state: case studies with Trivium and Trivia-SC , 2017, Des. Codes Cryptogr..