# Walsh permutation; bent functions

Walsh permutation |

Each bent function corresponds to a Walsh permutation, as shown in the following examples.

All bent functions in the same *sec* correspond to the same Walsh permutation.

Below are four examples how a Walsh permutation follows from a bent function. This is the meaning of the matrices:

*sec*matrix of the bent function- same matrix, XORs with top row indicated by circles
- only the XORs from last matrix
- rows complemented that had a 1 in the leftmost position (permuted binary Walsh matrix)
- permutation matrix corresponding to the permuted Walsh matrix

There are four big equivalence classes of 4-ary bent functions with Hamming weight 6.

## BEC 391

[edit | edit source]BEC 391 with only one SEC corresponds to the complemented bit permutation *wp*(14,13,11, 7):

## BEC 381

[edit | edit source]BEC 381 with 3 SECs corresponds to the bit permutations *wp*(2,1,8,4), *wp*(4,8,1,2) and *wp*(8,4,2,1).

They correspond to the simple permutations with the numbers 7, 16 and 23 - the three permutations of 4 elements that have two 2-cycles.

## BEC 382

[edit | edit source]BEC 382 with 12 SECs corresponds to 12 Walsh permutations with the following compression vectors:

4 8 9 6 6 9 1 2 |
8 4 10 5 10 5 2 1 |
2 9 8 6 6 1 9 4 |
8 12 2 3 12 4 3 1 |
2 5 10 4 10 1 8 5 |
4 12 3 2 12 8 1 3 |

The cubes of these permutations are the three bit permutations mentioned in the section above.

The cube of *wp*(12, 4, 3, 1) is *wp*(4, 8, 1, 2)

## BEC 383

[edit | edit source]BEC 383 with 12 SECs corresponds to 12 Walsh permutations with the following compression vectors:

2 13 10 6 14 1 9 5 |
10 13 2 3 12 4 11 5 |
6 5 11 4 10 9 8 7 |
4 12 11 6 14 9 1 3 |
6 13 3 2 12 8 9 7 |
8 12 10 7 14 5 3 1 |

The squares of these permutations are the complements of the bit permutations that correspond to the six transpositions of four elements.

The square of *wp*(12, 4, 11, 5) is *wp*(14, 11, 13, 7). That is the complement of *wp*(1, 4, 2, 8), which corresponds to the transposition of four elements that swaps the two in the middle.