Improved cost function in the design of Boolean functions satisfying multiple criteria

Kavut S., Yucel M.

PROGRESS IN CRYPTOLOGY -INDOCRYPT 2003, vol.2904, pp.121-134, 2003 (Peer-Reviewed Journal) identifier identifier

  • Publication Type: Article / Article
  • Volume: 2904
  • Publication Date: 2003
  • Doi Number: 10.1007/978-3-540-24582-7_9
  • Journal Indexes: Science Citation Index Expanded, Scopus, Compendex, EMBASE, MathSciNet, Philosopher's Index, zbMATH
  • Page Numbers: pp.121-134
  • Keywords: simulated annealing, bent Boolean functions, nonlinearity, Walsh-Hadamard transforms, autocorrelation, GLOBAL AVALANCHE CHARACTERISTICS, COVERING RADIUS, NONLINEARITY


We develop an improved cost function, to be used, in simulated annealing followed by hill-climbing to find Boolean functions satisfying multiple desirable criteria such as high nonlinearity, low autocorrelation, balancedness, and high algebraic degree. Using this cost function that does not necessitate experimental search for parameter tuning, the annealing-based algorithm reaches the desired function profiles more rapidly. Some Boolean functions of eight and nine variables have been found, which are unattained in the computer search based literature, in terms of joint optimization of nonlinearity and autocorrelation. Global characteristics of eight-variable Boolean functions generated by algebraic construction or computer search are compared with respect to the sum-of-squared-error in their squared spectra, which is also proportional to the sum-of-squared-errors in their autocorrelation function, the term 'error' denoting the deviation from bent function characteristics. Preliminary results consisting of cryptographically strong Boolean functions of nine, ten and eleven variables obtained using a three-stage optimization technique are also presented.