Chaotic transformation of a
vector

The suggested transformation transforms a vector of dimension **n** into
a "chaotic vector" of the same dimension, i.e. in a vector whose components
are permutated in pseudo-random way with an algorithm which generates a chaotic
succession of numbers.

This transformation is reversible.

If one describes a vector like a succession of n components **Ci** of value **Vi**. Each component has a position **Ai **=i.

The original vector can be written:

C1, Ci...,Cnof valuesV1, Vi,Vn- i { 1.. n }

The transformation works on the position of the successive components of the
vector: the series of the positions undergoes a permutation which delocalizes
the components of the original vector while preserving their value.

For each component **Ci** of address **Ai**,
one calculates a new address **A'i** which corresponds
to the position of this component in the transformed vector.

A'i = i * b mod (n+1)

- i { 1..n }
- b , (n+1) incommensurable numbers

If one calculates the address **A'i** starting
from the address **A' i-1**, the series of **A'i**
can be written:

A'i = (A' i-1 + b) mod (n+1)- A' 0 = 0
- i{ 1.. n }
- b , (n+1) incommensurable numbers

It is noticed that this series is a particular form of

Ai = (a * Ai-1 + b) mod (n+1)- a = 1
- i { 1.. n }
- b , (n+1) incommensurable numbers

which generates the succession of the numbers
of **1** to **n** in a chaotic order [DEW87]. given **n**, this
series is different for each value of parameter **b**, but recurrences appear
when the number of possible permutations is reached (n-1).

**A- Value of b lower than n+1: b = 3** and **n = 6**,

the transformed vector

C'1, C'2, C'3, C'4, C'5, C'6

is obtained by permutation of the components of the original vector:

A' 1 = (1*3) mod 7 = 3

A' 2 = (2*3) mod 7 = 6

A' 3 = (3*3) mod 7 = 9-7 = 2

A' 4 = (4*3) mod 7 = 12-7 = 5

A' 5 = (5*3) mod 7 = 15-7*2 = 1

A' 6 = (6*3) mod 7 = 18-7*2 = 4

**B- value of b higher than n+1: b = 23** and **n = 6**,

A' 1 = (1*23) mod 7 = 23-7*3 = 2

A' 2 = (2*23) mod 7 = 46-7*6 = 4

A' 3 = (3*23) mod 7 = 69-7*9 = 6

A' 4 = (4*23) mod 7 = 92-7*13 = 1

A' 5 = (5*23) mod 7 = 115-7*16 = 3

A' 6 = (6*23) mod 7 = 138-7*19 = 5

**let us visualize** this last transformation:

This vectorial transformation is **a permutation** which is characterized
by **a matrix of permutation **n x n (P) built starting from the chaotic
sequence corresponding with a parameter b given.

example with **b = 3**:

The transformation is bi-univocal and the reverse transformation is obtained
while multiplying the vector transformed by the opposite matrix.

[DEW87] Dewdney A., Explorez le monde étrange du chaos, Récréations informatiques, in Pour la Science , N 119, Sept 87, p 13-16

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