# Introduction

A Boolean function of n variables is a function ƒ: Z2n → Z2. Such functions are used to construct and analyse cryptographic systems.

Equivalence classes of Boolean functions can be defined with respect to various symmetries:

A) Variable permutations
ƒ(x0, x1, …, xn-1) = ƒ(xp(0), xp(1), …, xp(n-1)).
B) Variable complementations
ƒ(x) = ƒ(x + a).
C) Affine offset
ƒ(x) = ƒ(x) + bx + c.
D) Linear mapping
ƒ(x) = ƒ(xA), where A is an invertible binary matrix.

Different properties of Boolean functions are equivalent with respect to different symmetries:

Properties equivalent with respect to A, B, C, and D
• Algebraic degree.
• Non-linearity.
• Absolute distribution of the coefficients of Walsh spectrum and autocorrelation spectrum. (Order and signs may change.)
Properties equivalent with respect to A, B, and C.
• Maximum satisfied degree of the propagation criterion.
Properties equivalent with respect to A.
• Balancedness.
• Maximum order of correlation immunity.
• Maximum order of resilience.

All equivalence classes of Boolean functions of up to 6 variables with respect to A, B, C, and D are available from [1]. Propagation criterion and correlation immunity of Boolean functions of up to 6 variables have been analysed in [2].

^ TOP

# Tables

Number of equivalence classes of Boolean functions with respect to symmetries A, B, C, and D. (These numbers are also found as sequence A001289 in the On-Line Encyclopedia of Integer Sequences.)

n123456
#123848150,357

Number of equivalence classes of Boolean functions with respect to symmetries A, B, and C.

n12345
#1253922,442

The number of equivalence classes of Boolean functions with 6 variables and degree less than or equal to 3 is 851,520.

^ TOP

# File formats

The equivalence classes with respect to A, B, and C are listed in a file consisting of records of the following format:

• number of variables
• algebraic degree
• non-linearity
• maximum satisfied degree of the propagation criterion
• absolute distribution of the coefficients of the Walsh spectrum
• Example: "(0,24)(8,7)(24,1)" means that there are 24 coefficients with value 0, 7 coefficients with abs. value 8, and 1 coefficient with abs. value 24.
• absolute distribution of the coefficients of the autocorrelation spectrum
• a Boolean function from the class
• The Boolean functions are given in algebraic normal form notation. Example: "03,13,23" is the function "x0x3+x1x3+x2x3".

^ TOP

# Files

## Equivalence classes with respect to A,B,C

(n ≤ 5, and n=6 with degree ≤ 3)