Diverses méthodes de construction de

générateurs de nombres pseudo

aléatoires à partir d’itérations de

fonctions non-linéaires ou linéaires par

morceaux

René Lozi

Laboratoire J. A. Dieudonné,

UMR du CNRS 7351

rlozi@unice.fr

True Pseudo-Random Numbers vs Pseudo-

Random Numbers and Chaotic Numbers

“True” Random Numbers are produced generally by physical devices

such as presented in the next slide.

However they are not “controllable”, that means one cannot

synchronize two devices generating Random Numbers. Therefore

they are not useful for cryptography for which the coding and

decoding processes are linked by keys (either secret or public).

Pseudo-Random Numbers (as those generated by function Rand in

your computer) depend on an initial seed (for example the time in

microsecond of the internal clock of your computer used as a

guess). Using the same seed gives the same sequence of Pseudo

Random Numbers.

Chaotic Pseudo-Random Numbers are built from Chaotic Numbers,

they are equipped with more astute parameters than only one initial

seed. They are more fitted for cryptography based chaos. Those

parameters can be used as keys.

Example of physical random number generator

QUANTIS is a physical

random number generator

exploiting an elementary

quantum optics process.

One needs to use Quantis

in connection with a

computer or server.

The product exists in three

versions compatible with

most platforms:

• USB device – random stream of 4Mbits/sec

• PCI Express (PCIe) board – random stream of 4Mbits/sec

• PCI board – random stream of 4Mbits/sec and 16Mbits/sec

Pseudo-Random Numbers

Generators (PRNG)

They are many kinds of generators:

- Linear Congruential generator

- Blum Blum Shub (B.B.S.)

- Mersenne Twister

1 modn nx ( ax c ) m

2

1n nx x mod M

The Blum Blum Shub PRNG

Blum Blum Shub (B.B.S.) is a Pseudo Random Number

Generator proposed in 1986 by Lenore Blum, Manuel

Blum and Michael Shub.

It takes the form:

where is the product of

two large primes.

At each step of the algorithm the output is derived from

The output is commonly either the bit parity of or one or more of

least significant bits of

2

1n nx x mod M M pq

1nx

1nx

1nx

The Blum Blum Shub PRNG: conditions

1- The seed should be an integer that is co-prime

with ,i.e. are not factors of , and

not 0 or 1.

2- the two primes should both be congruent

to 3 (mod 4) and should be

small,

where is the Euler function which is, in this case, the

number of integers in the range for

which the greater common divisor of is

M0x

p and q 0x

p and q1 1gcd( ( p ), (q ))

1 k n k

1gcd( n,k ) k

The dawn of chaotic

dynamical systems

(from early beginning to nowadays)

The dawn of chaotic iterations (I)

The study of nonlinear dynamics is relatively recent

with respect to the long historical development of the

early mathematics since the Egyptian and the Greek

civilizations. The beginning of this study can be

traced to the phenomenal work of Henri Poincaré.

The Poincaré map being an essential tool linking

differential equations and mappings. Henri Poincaré

(1854-1912) Concerning iterations theory,

one has to include in this field

of research the pioneer works

of Gaston Julia and Pierre Fatou

related to one-dimensional maps

with a complex variable, near

a century ago.

Pierre Fatou Gaston Julia

1878-1929 1893-1978

The dawn of chaotic iterations (II)

In France Igor Gumosky and Christian Mira began

their mathematical researches in 1958. They

produced a considerable work on the matter (theory

of boxes in the boxes for example). Among their

discoveries one can emphasize on their family of

attractors from an aesthetic point of view (of course

Christian Mira it is only a microscopic point of view of what they

haveproduced)

The Gumowski-Mira attractor:

Christian Mira

is sensitive to slight changes of parameters a and b

The dawn of chaotic iterations (III)

a = - 0.918, b = 0.9

a = - 0.93333, b = 0.92768

The dawn of chaotic iterations (IV)

In Japan the Hayashi’s School (with disciples like Ikeda, Ueda and Kawakami)

in the same period, were motivated by applications to electric and electronic

circuits. Mappings were used as models of behavior of electric circuits.

The Ikeda attractor (1980):

has a chaotic attractor when u 0.6

u = 8.6

u = 8,9

The dawn of chaotic iterations (V)

In the last 50 years long history of chaotic iterations leading to the new

concept of strange attractors, and corresponding chaotic differential

systems, one can mention few important dates:

Sharkovsky order Lorenz attractor Rössler attractor Hénon map

1962 1963 1976 1976

Belykh map 1976 Chua attractor 1983 Chen attractor 1999

Two-Dimensional discrete dynamical systems: Hénon

mapping (1976)

Simplest model of Poincaré map of Lorenz equation.

Associated dynamical system

with initial value:

Linearized version in 1978

a 1.4, b 0.3 Ha,b

2x y 1 ax:y bx

La,b

x y 1 a x:

y bx

2 2

a,bH :

2n 1 n n

n 1 n

x y 1 ax

y bx

a 1.7, b 0.5

0

0

x

y

Fractal structure of the Hénon attractor

First zoom Second magnification

Fractal structure of the Hénon attractor

Structure of Cantor set

+ Sensitivity to initial

conditions

= strange attractor

Sensitivity to initial

conditions is assessed by

Lyapunov exponents

Third zoom

Chaotic Pseudo-Random Number Generators

Mastering high quality

randomness via chaos

theory

One-Dimensional discrete dynamical systems: Logistic

and Tent Map

Maps of some interval included in the real line

Logistic map symmetric tent map

Associated dynamical system Associated dynamical system

n 1x rx ( 1 x )n n

f ( x ) rx( 1 x )r

f ( x ) 1 2 x

n 1x 1 2 xn

f :

f : 0,1 0,1 f : 1,1 1,1

Symmetric tent map

Invariant measure

= Lebesgue measure

However numerical instability

leads to the collapse of

solutions to the unstable

fixed point x = -1

G. Yuan & J. A. Yorke, Collapsing of chaos in one dimensional maps, Physica D, 136,18-30 (2000).

: 1,1 1,1f

2a

af ( x ) 1 a x

-1

-0,5

0

0,5

1

-1,0 -0,5 0,0 0,5 1,0

symmetric tent map

Density of iterated values of the Logistic map

0

0,5

1

0,0 0,3 0,5 0,8 1,0

logistic map on the unit square

1 4 1n n nx x x

0

1

2

3

4

5

0,0 0,3 0,5 0,8 1,0

Invariant measure of the logistic map

1( )

(1 )P x

x x

The route from chaos to pseudo-randomness via ultra-

weak coupling, and chaotic or mixing undersampling

Step 1: Ultra-weak coupling of 1-D maps

Ultra-weak coupling means

for floating points or for double precision numbers

Ultra-weak coupling is efficient in order to restore numerically the chaotic properties of chaotic mappings, avoiding any numerical collapse

710i

f ( x ) 1 2 x n 1x 1 2 xn

1410i

1 1 2 3 4

n 1 1 n 1 n 1 n 1 n

2 1 2 3 4

n 1 2 n 2 n 2 n 2 n

3 1 2 3 4

n 1 3 n 3 n 3 n 3 n

4 1 2 3 4

n 1 4 n 4 n 4 n 4 n

x (1 3ε ) f ( x ) ε f ( x ) ε f ( x ) ε f ( x )

x ε f ( x ) (1 3ε ) f ( x ) ε f ( x ) ε f ( x )

x ε f ( x ) ε f ( x ) (1 3ε ) f ( x ) ε f ( x )

x ε f ( x ) ε f ( x ) ε f ( x ) (1 3ε ) f ( x )

The main criteria for CPRNG robustness

Approximate distribution of iterates

We define several function errors, which assess the between uniform distribution and discrete repartition of iterates in boxes. In the case of 1-D mapping, one can assess, the repartition of the iterates on the Interval, but not only: the space of delay must also be considered.

Such functions depend on the number of boxes

and the number of iterates

In the case of p-Dimensional mapping, much more functions are needed

First numerical results

First numerical results

Variation of the error vs the number of iterations and the number of boxes

Variation of the error vs the initial values

Correlation between variables

Step 2: Chaotic and mixing under sampling

1 4

1 2

2 4

2 3

3 4

3

,

,

,1

n n

q n n

n n

x iff x T T

x x iff x T T

x iff x T

Example in 4-D: Let be three thresholds

instead of using directly the coupled sequences

One mixes and samples those sequences using the fourth one:

using

In order to obtain:

which are pseudo-random. ,,,,,, 1210 qq xxxxx

-1 < T1 < T2 < T3 < 1

,,,,,, 1

1

11

2

1

1

1

0 nn xxxxx

4 4 4 4 4

0 1 2 1n nx , x , x , , x , x ,

2 2 2 2 2

0 1 2 n n 1x , x , x , , x , x , and

,,,,,, 3

1

33

2

3

1

3

0 nn xxxxx

Correlation between variables

Window of randomness versus

Geometric undersampling

Step 1: Ring coupling of several tent maps Instead of using one single tent maps , we use simultaneously

several (up to 10 or 20) tent maps coupled in a ring way.

Moreover we restrain de new p-dimensional map to the torus:

: 1,1 1,1f

1,1p

Ring coupling of p tent maps

In the p-dimensional case the mapping is defined by:

In order to confine the vector Xn+1 on

the torus , one modifies the

components in the following way:

1 1 2

n 1 n 1 n

m m m 1

n 1 n m n

p 1 p 1 p

n 1 n p 1 n

p p 1

n 1 n p n

x 1 2 x k x

x 1 2 x k x

x 1 2 x k x

x 1 2 x k x

j

n 1

j

n 1

if ( x 1) add 2

if ( x 1) substract 2

1,1p

The NIST tests

NIST Test for p=10

Example: K1, K2, K3, K4, K5, K6, K7, K8, K9, K10 = 1

(joint work with Ina Taralova and Andrea Espinel-Rojas)

Conclusion: each stream generates pseudo-random

numbers, moreover these streams are uncorrelated.

Chaotic multi-stream pseudorandom number generators

(Cms-PRNG)

Using together ring coupling and full ultra-weak coupling,

it is possible to build a Pseudo Random Number

Generator with a huge number of keys.

1, j

p p1 2 jn 1 1 n 1, j n

j 3 j 3

p pm m m 1 jn 1 n m m, j n m, j n

j 1, j m;m 1 j 1, j m;m 1

p 2 p 2p 1 p 1 p j

n p 1 p 1, j n p 1, j nn 1j 1 j 1

1x 1 2 x k 1 x xn

x 1 2 x k 1 x x

x 1 2 x k 1 x x

p 1 p 1p p 1 j

n p p , j n p , j nn 1j 2 j 2

x 1 2 x k 1 x x

Step 2: Ring coupling of 2 tent maps

Example with 2 coupled symmetric tent maps:

the vector Xn+1 is confined on the torus identified to a square.

The coefficients ki are set to +1

with

It is possible to define critical lines forming a partition of the square.

1 1 2

n 1 n n

2 2 1

n 1 n n

x 1 2 x x

x 1 2 x x

j

n 1

j

n 1

if ( x 1) add 2

if ( x 1) substract 2

2

1,1

Invariant partition of the square

First quadrant (I) Second quadrant (II)

Fourth quadrant (IV) Third quadrant (III)

Exact computation of invariant measure

First quadrant (I) Image of the first quadrant f(I)

Comparison with numerical iterations

Geometric subsampling

We first select the iterated points belonging to a subsquare

of the lozenge

We then enlarge it to the initial square

Exploring new topologies

of network of coupled

chaotic maps

Tent-Logistic map

We introduce a combined Tent-Logistic map:

When used in more than one dimension, map can be

considered as a two variable map:

TL

2 2f ( x ) TL ( x ) L ( x ) T ( x ) x x ( x x )

1 2 1 2 2( ) ( ) ( ) ( )TL ( x ,x ) ( x ( x ) )

TL

Ring coupling of Tent with Tent-Logistic maps

Hence it possible to define a mapping:

where and the coefficients ki are set to 1

In order to hold dynamics

in the torus we use the injection:

p pM : J J

p

Rp pp

J 1,1

j

n 1

j

n 1

if ( x 1) add 2

if ( x 1) substract 2

(1) 1 (1) (2)(1) (1)

1

(2) 2 (2) (3)(2) (2)

1

( ) ( ) (p) (p) (1)1

( ) ( , )

( ) ( , )

( ) ( , )

n n nn n

n n nn n

p

p p pn n n n n

T x k TL x xx x

T x k TL x xx x

M

x x T x k TL x x

2-Dimensional topologies

Among four choices of we select two cases:

1/ The ring coupling non-alternative map :

2/ the simple coupling alternative:

( 1 ) ( 1 ) ( ) ( ) 2x 1 x ( x ( x ) )n n nn 1( 1 ) ( 2 )RCTTL ( x , x )

n n ( 2 ) ( 2

2

) ( 1 ) ( 2 ) 2x 1 x ( x ( x ) )n n n

1

n 1

( 1 ) ( 1 ) ( ) ( ) 2x 1 x ( x ( x ) )n n nn 1( 1 ) ( 2 )SCTTL ( x , x )

n n ( 2 ) ( 2

1

) ( 1 ) ( 2 ) 2x 1 x ( x ( x ) )n n n

2

n 1

M2

The main criteria for CPRNG robustness

Bifurcation diagram of 2-D new map non alternative

RCTTL

Left: Largest Lyapunov exponent of non alternative

Right: Largest Lyapunov exponent of alternative

When LLE > 0 there is chaos

RCTTL

SCTTL

The parameter μ is set to 2

Left: Phase space behaviour of non alternative,

Plot of 20,000 points.

Right: Phase space behaviour of non alternative,

Plot of 20,000 points.

RCTTL

2

SCTTL

2

A new 2-D chaotic PRNG

In order to improve the previous topologies, we define a

new map with = 2

With a new injection mechanism which fits better the Torus

( 1 ) ( 2 )

n n

( 1 ) ( 2 ) ( 1 )2x 1 2( x ) ) 2 xn nn 1SCMTTL (

2 ( 2 ) ( 2 ) ( 1 ) ( 2 )2x 1 2( x ) 2( x x )n n nn 1

x ,x )

1

1

2

1

2

1

1 2

1 2

1 2

( )

n

( )

n

( )

n

if ( x ) then substract

if ( x ) then add

if ( x ) then substract

2 21,1 R

Injection mechanism of alternative map

SCMTTL

2

Left: Approximate density function of alternative

map, on the plane.

Right: Approximate density function of alternative

map, on the phase delay plane

SCMTTL

2

SCMTTL

2

( 1 ) ( 2 )( x , x )

( 1 ) ( 1 )( x , x )

n n 1

Approximate density function of alternative map

SCMTTL

2

NIST Test for

Results for x(1), the same results are obtained for x(2)

Conclusion: each stream generates pseudo-random

numbers, these streams are uncorrelated

SCMTTL

2

p-Dimensional topologies

We generalize the coupling non-alternative map from

dimension 2:

to dimension p:

( 1 ) ( 1 ) ( 2 ) ( 1 ) 2x 1 2 x 2( x ( x ) )n n nn 1

( 2 ) ( 2 ) ( 3 ) ( 2 ) 2RC,pD ( 1 ) ( 2 ) ( p ) x 1 2 x 2( x ( x ) )n n nn 1TTL ( x , x , , x )2 n n n

( p ) ( p ) ( 1 ) ( p ) 2x 1 2 x 2( x ( x ) )n n nn 1

( 1 ) ( 1 ) ( 2 ) ( 1 ) 2x 1 2 x 2( x ( x ) )n n nn 1( 1 ) ( 2 )RC,2DTTL ( x , x )

2 n n ( 2 ) ( 2 ) ( 1 ) ( 2 ) 2x 1 2 x 2( x ( x ) )n n nn 1

NIST Test for

Results for x(1), the same results are obtained for x(2), x(3),

x(4),

Conclusion: each stream generates pseudo-random

numbers which are uncorrelated

RC,4DTTL

2

Other numerical experiments using

multi-core processor

Equal density test

Graph of the error versus equal density on in dimension

p = 2 to 5, (horizontal axis, logarithmic value of the number of random

numbers).

pp1,1 R

These results show that the pace of computation is very high.

When is the mapping tested, and the machine used is

a laptop computer with a Core i7 4980HQ processor with 8

logical cores, computing 1011 iterates with five parallel streams

of PRNs leads to around 2 billion PRNs being produced per

second.

Since these PRNs are computed in the standard double precision

format, it is possible to extract from each 50 random bits (the

size of the mantissa being 52 bits for a double precision floating-

point number in standard IEEE-754). Therefore,

can produce 100 billion random bits per second, an incredible

pace! With a machine with 4 Intel Xeon E7-4870 processors

having a total of 80 logical cores, the computation is twice as

fast, producing 200 billion random bits per second.

RC,5DTTL

2

RC,5DTTL

2

Merci de votre attention