New Possibilities for Cellular Automata in Cryptography · 2016-02-22 · • Debasis, Das and Abhishek Ray. A Parallel Encryption Algorithm for Block Ciphers Based on Reversible

New Possibilities for

Cellular Automata in

Cryptography

Mauro Tardivo Filho

Marco A. A. Henriques

Faculty of Electrical and Computer Engineering

University of Campinas

Sao Paulo - Brazil

2

Overview

1.History

2.Cellular Automata

3.Chaotic Cellular Automata

4.Application of Chaotic Cellular Automata in Cryptography

5. Parallelism in Cellular Automata

1.Mechanisms of parallelism

2.Results

6.New results

7.Conclusion

3

History

• Universal copier and constructor (1940s):

• Stanislaw Ulam: growth of crystals;

• John von Neumann: self-replicating systems

in robotics.

• John Conway: Game of Life (1970s)

• Stephen Wolfram: analyzed the behavior and

complexity of cellular automata (1983).

4

Elementary Cellular

Automata (CA)

• Discrete dynamic system

• neighborhood radius r

D=1,

r=1,

c= 2

t

t+1

t+2

N

r

= 0

= 1

States (s) ...

...

...

...

...

...

• s states

• synchronous evolution

depending on rule R

5

CA Rules

• One-dimensional CA:

• next state of cell i:

• Example: Rule 3010 = 1E16 = 000111102

Tim

e

6

CA evolution

Source: Wolfram, “A New Kind of Science”, 2002

• Evolution depends strongly on initial

conditions and rule used

Tim

e

1-D

2-D

7

Chaotic Behaviour

• Some rules cause a chaotic behaviour => Chaotic CA

(CCA)

• The difference between the evolutions from two

similar initial states increases with respect to time;

• consequence: great dependence on initial conditions.

• future states cannot be predicted unless doing a state

by state calculation;

• cost to obtain previous states can be made very high,

increasing the number of CA cells and/or the radius r.

8

Chaotic CA Evolution

• Very similar conditions can led to very different

final states

Tim

e

9

Computational Unpredictability

and Irreversibility

Prediction:

impossible

Reversion:

costly

Tim

e

1

0

Applications of CCA in

Cryptography

• Random Number Generators

• Hash Algorithms

Input/Seed Input Input

Output Output Output

k-iteratio

ns

1st generation 2nd generation nth generation

1

1

Applications of CCA in

Cryptography

• Example: Vernam cipher key generator

Seed

k-iteratio

ns

Plain Text: Hello World 010010000110010101101100011011000110111100100

0000101011101101111011100100110110001100100

111100010000000010101010101010101001110100101

0011110011111110001001101000001000011010011

Cipher Text:

= 101110010110010111000110110001101111001000001

0011011000010011110010001100111110010110111

9aHew(d0=M$

1

2

CA robustness

• Bao (2003) showed vulnerabilities in CA

cryptosystems: they can be predictable.

• Solutions:

• use different rules for differente cells

(Hybrid CA);

• increase number of cells (N), neighborhood

radius (r) or number of iterations (k);

• use rules with good chaotic behaviour.

1

3

What is good,

can become better?

1

4

Parallelism in CA • The evolution of CA can be calculated using

parallel processing:

• take advantage of multicore processors;

• increase performance

N=20, r = 1, 4 cores

.

.

.

.

.

.

.

.

.

.

.

.

1

4

1

5

Parallel CA Experiments

• Large CA simulated using parallelism

• number of cells: 27 <= N <= 218 (128-

256k bits)

• cores: 2

• rule: 3010 = 1E16

• iterations: k = 2N

• parallel C libraries:

• OpenMP

• PThreads

1

6

Results (1)

Runtime as a function of the input size (2 cores)

time

decreases

1

7

Results (2) Speedup as a function of the (log2)

input size (2 cores)

Speedupmax=1.65 Gain over

single-

processing

1

8

New Results

• Impact of radius r on parallel processing of CAs

• Algorithm optimization

1

9

Impact of radius r on

processing parallel CA • Each of p cores, will

read ((N/p) + 2r) cells

at each iteration.

• The cells on the gray

area are shared by two

cores for reading, but

not for writing ==>

synchronization is

easier.

t

t+1

t+2

r=1

r r

...

...

...

...

...

...

2

0

Problem using larger

radius r

• Problems:

• Memory access

bottleneck

• Cores will need

more time to

synchronize

t

t+1

t+2

r=4

r r

...

...

...

...

...

...

2

1

Algorithm Optimization

• A pseudo-random number

(bit) generator can be obtained

from the central cell during the

CA evolution.

• In this case, fewer cell states

need to be calculated => black

areas cells can be ignored.

• Number of states to update:

• Traditional: N*k

• Optimized: N*(k - h/2)

N

k-iteratio

ns

h

2

2

Computational effort

reduction Reduction of steps as a function of number of

iterations k and radius r

2

3

Conclusions • CAs have good characteristics to be used as random number

generators and hash functions.

• However, they need adequate sizes and good set of rules.

• Popularity of multicore processors (even on mobile devices) is

increasing the adoption of parallel processing.

• CA can be easily implemented and scaled in parallel

architectures.

• However, experiments indicate that the choice of

appropriate mapping and programming tools is crucial to

the success of a parallel implementation.

• Some optimizations can be made to reduce the volume of

calculation needed and the actual reduction depends strongly

on the size and other CA parameters.

2

4

Future Works • New research efforts are needed to better

understand:

• the strength of CA with larger

neighborhoods (radius);

• the real benefits that can be obtained from

parallel processing techniques in multicore

enviroments;

• the impacts of a CA parallel implementation

on the overall system security.

2

5

Thank You!

Gracias! Obrigado!

• [email protected]

• [email protected]

• twitter: @maurotfilho

• about.me/maurotfilho

2

6

Boundary Conditions

• Null Neighbourhood

• Cyclic Neighbourhood

= 0 = 1

...

...

... N

At the extremes of each line, where the

neighboring cells are not physically adjacent, we

use one of the following approaches:

t

t+1

t+2

t

t+1

2

7

• Szaban et. al (2006) implemented a Genetic

Algorithm (GA) which found sets of rules

for CA with good non-linearity and

randomness, using neighborhoods of radii 1

and 2.

• As a future work, larger neighborhoods can

be explored to possible find better rules.

CA Important Facts

2

8

References • Petre Anghelescu, Silviu Ionita, and Ionel Bostan. Design of programmable cellular automata based cipher scheme. 2009

World Congress on Nature & Biologically Inspired Computing (NaBIC), pages 187–192, 2009.

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• Debasis, Das and Abhishek Ray. A Parallel Encryption Algorithm for Block Ciphers Based on Reversible Programmable Cellular Automata. Journal of Computer Science, 1(1):82– 90, 2010.

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• The OpenMP API specification for parallel programming. http://openmp.org/wp/openmp- specifications/.

• Tommaso Toffoli and Norman Margolus. Invertible cellular automata: A review. Physica D 45, pages 229–253, 1990.

• Stanislaw Ulam. Random process and transformations. Proceedings of the International Congress on Mathematics, Vol. 2 (1952):264– 275, 1950.

• John von Neumann. The general and logical theory of automata. Collected Works, 5:288, 1963.

• John von Neumann. Theory of Self- Reproducing Automata. University of Illinois, Urbana, 1966.

• Stephen Wolfram. Theory and applications of cellular automata. Rev. Mod. Physica, 55(601), 1983.

• Stephen Wolfram. Universality and complexity in cellular automata. Physica D10, 1984.

• Stephen Wolfram. Random sequence generation by cellular automata. Adv Appl Math, 7:123, 1986.

• Stephen Wolfram. A New Kind of Science. Wolfram Media, Inc, 2002.