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A quantum algorithm to approximate the linear structures of Boolean functions

Published online by Cambridge University Press:  09 February 2016

HONGWEI LI  and
LI YANG
HONGWEI LI
Affiliation:
State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China Email: hwli428@sina.com School of Mathematics and Statistics, Henan Institute of Education, Zhengzhou 450046, Henan, China Data Assurance and Communication Security Research Center, Chinese Academy of Sciences, Beijing 100093, China Email: yangli@iie.ac.cn University of Chinese Academy of Sciences, Beijing 100049, China
LI YANG*
Affiliation:
State Key Laboratory of Information Security, Institute of Information Engineering, Chinese Academy of Sciences, Beijing 100093, China Email: hwli428@sina.com Data Assurance and Communication Security Research Center, Chinese Academy of Sciences, Beijing 100093, China Email: yangli@iie.ac.cn
*
‖Corresponding author.
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Abstract

A quantum algorithm to determine approximations of linear structures of Boolean functions is presented and analysed. Similar results have already been published (see Simon's algorithm) but only for some promise versions of the problem, and it has been shown that no exponential quantum speedup can be obtained for the general (no promise) version of the problem. In this paper, no additional promise assumptions are made. The approach presented is based on the method used in the Bernstein–Vazirani algorithm to identify linear Boolean functions and on ideas from Simon's period finding algorithm. A proper combination of these two approaches results here to a polynomial-time approximation to the linear structures set. Specifically, we show how the accuracy of the approximation with high probability changes according to the running time of the algorithm. Moreover, we show that the time required for the linear structure determine problem with high success probability is related to so called relative differential uniformity δf of a Boolean function f. Smaller differential uniformity is, shorter time is needed.

Type
Paper
Information
Mathematical Structures in Computer Science , Volume 28 , Issue 1 , January 2018 , pp. 1 - 13
Copyright
Copyright © Cambridge University Press 2016 

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