New Possibilities for Cellular Automata in Cryptography

Transcription

1 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 2

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). 3

4 Elementary Cellular Automata (CA) Discrete dynamic system neighborhood radius r s states synchronous evolution depending on rule R r States (s) = 0 t t+1 D=1, r=1, = 1 t+2 4 N c= 2

5 CA Rules One-dimensional CA: next state of cell i: Example: Rule 3010 = 1E16 = Time 5

6 CA evolution Evolution depends strongly on initial conditions and rule used 1-D Time 2-D Source: Wolfram, A New 6 Kind of Science, 2002

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. 7

8 Chaotic CA Evolution Very similar conditions can led to very different final states Time 8

9 Computational Unpredictability and Irreversibility Prediction: impossible Time Reversion: costly 9

10 Applications of CCA in Cryptography Random Number Generators Hash Algorithms Input/Seed Input Input k-iterations Output Output Output 1 st generation 2 nd generation n th generation 1

11 Applications of CCA in Cryptography Example: Vernam cipher key generator Seed Plain Text: Hello World k-iterations = Cipher Text: 1 9aHew(d0=M$

12 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

13 What is good, can become better? 1

14 Parallelism in CA The evolution of CA can be calculated using parallel processing: take advantage of multicore processors; increase performance N=20, r = 11, 4 cores 1

15 Parallel CA Experiments Large CA simulated using parallelism number of cells: 2 7 <= N <= 2 18 ( k bits) cores: 2 rule: 3010 = 1E16 iterations: k = 2N parallel C libraries: OpenMP PThreads 1

16 Results (1) Runtime as a function of the input size (2 cores) time decreases 1

17 Results (2) Speedup as a function of the (log2) input size (2 cores) Speedupmax=1.65 Gain over singleprocessing 1

18 New Results Impact of radius r on parallel processing of CAs Algorithm optimization 1

19 Impact of radius r on processing parallel CA Each of p cores, will read ((N/p) + 2r) cells at each iteration. r r The cells on the gray area are shared by two t+1 t cores for reading, but not for writing ==> t+2 synchronization is easier. r=1 1

20 Problem using larger radius r Problems: Memory access bottleneck t r r Cores will need more time to t+1 t+2 synchronize r=4 2

21 Algorithm Optimization A pseudo-random number (bit) generator can be obtained from the central cell during the CA evolution. N 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) k-iterations h 2

22 Computational effort reduction Reduction of steps as a function of number of iterations k and radius r 2

23 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

24 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

25 Thank You! Gracias! Obrigado! about.me/maurotfilho 2

26 Boundary Conditions At the extremes of each line, where the neighboring cells are not physically adjacent, we use one of the following approaches: Null Neighbourhood t t+1 t+2 Cyclic Neighbourhood t N t+1 = 0 = 1 2

27 CA Important Facts 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. 2

28 References Petre Anghelescu, Silviu Ionita, and Ionel Bostan. Design of programmable cellular automata based cipher scheme World Congress on Nature & Biologically Inspired Computing (NaBIC), pages , Blaise Barney. Posix threads programming. 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, Martin Gardner. The fantastic combinations of John Conway s new solitaire game "life". Scientific American, 1(223): , Juan Pedro Hecht. Autómatas celulares caóticos en la generación de funciones. IV Congreso Iberoamericano de Seguridad Informática CIBSI 07, pages , K J Jegadish Kumar, K Chenna Kesava, and S Salivahanan. Novel and Efficient Cellular Automata based Symmetric Key Encryption Algorithm for Wireless Sensor Networks. International Journal, 13(4), T. G. Mattos and J. G. Moreira. Universality classes of chaotic cellular automata. Brazilian Journal of Physics, 34(2a): , June S.Nandi, B.K.Kar, and P. Pal Chaudhuri. Theory and applications of cellular automata in cryptography. IEEE Transactions on Computers, 43(12): , The OpenMP API specification for parallel programming. specifications/. Tommaso Toffoli and Norman Margolus. Invertible cellular automata: A review. Physica D 45, pages , Stanislaw Ulam. Random process and transformations. Proceedings of the International Congress on Mathematics, Vol. 2 (1952): , John von Neumann. The general and logical theory of automata. Collected Works, 5:288, John von Neumann. Theory of Self- Reproducing Automata. University of Illinois, Urbana, Stephen Wolfram. Theory and applications of cellular automata. Rev. Mod. Physica, 55(601), Stephen Wolfram. Universality and complexity in cellular automata. Physica D10, Stephen Wolfram. Random sequence generation by cellular automata. Adv Appl Math, 7:123, Stephen Wolfram. A New Kind of Science. Wolfram Media, Inc,