Abstract
We describe a new random number generator, RPGM, which is based on the cryptographic system PGM invented by Magliveras in 1976 and subsequently studied by Magliveras and Surkan [10]. PGM relies on a certain method of machine representation for permutation groups. This method allows for encryption and decryption algorithms based on a space-efficient data structure which is called a logarithmic signature for the group. The efficacy of RPGM is studied by means of an extensive analysis of generated data of 100,000 numbers using the Mathieu groupM 24 in its 5-transitive representation on 24 points.
Sunto
Descriviamo un nuovo generatore di numeri a caso (RPGM), basato sul sistema crittografico PGM inventato da Magliveras nel 1976 e successivamente studiato da Magliveras e Surkan [10]. PGM si fonda su un certo metodo di rappresentazione in un computer di un gruppo di permutazioni. Questo metodo dà luogo ad algoritmi di incrittazione e decrittazione basati su una struttura di dati efficienti, chiamata segnatura logaritmica del gruppo. L'efficacia di RPGM è studiata ricorrendo ad un'ampia analisi dei dati generati di 100.000 numeri, ottenuti usando il gruppo di Mathieu M24 nella sua rappresentazione 5-transitiva di grado 24.
Similar content being viewed by others
Bibliography
Berlekamp E. R., «Algebraic coding theory», McGraw-Hill, New York, 1968.
Bright H. S. andEnison R. L., «Quasi-Random Number sequences from a Long-Period TLP Generator with Remarks on Application to Cryptography»,ACM Computing Surveys, Vol. 11, no. 4, December 1979, pp. 358–370.
Butler G., «The Schreier Algorithm for Matrix Groups», Symposium on Symbolic and Algebraic Computation,SYSMAC '76, 1976, p. 167.
Cannon John J., «On Determining the Order of a Group»,Proceedings of the 1976 ACM Symposium on Symbolic and Algebraic Computation, Yorktown Heights, New York, 1976. Also:SIGSAM Bull., Vol. 10, No. 3, 1976, p. 5.
Felsch, V., «Programs for Permutation Groups», Todd-Coxeter,Defining Reations Survey, Permutations (Actes Colloq., University Rene-Descartes, Paris, 1972), Gauthier-Villars, Paris, 1974, pp. 241–250.
Friedman W. F., «Cryptology»,Encyclopedia Britannica, Vol. 6, 1967, pp. 844–851.
Golomb S. W., «Shift Register Sequences», Holden-Day, San Francisco, California, 1967.
Hall M., «The Theory of Groups», MacMillan, 1959.
Knuth D. E., «The Art of Computer Programming», Vol. 2,Seminumerical Algorithms, Second Edition, Addison-Wesely, Reading, Mass., 1981, pp. 38–75.
Magliveras S. S. andSurkan A. J., «A Cryptosystem from Lograrithmic Signatures of Finite Groups», to appear in the Proceedings of the 29th Midwest Symposium on Circuits and Systems, Elsevier Publ. Co., 1986.
Morris R., Sloane N. J. A. andWyner A. D., «Assessment of the National Bureau of Standards Proposed Federal Data Encryption Standard»,Cryptologia, Vol. 1, No. 3, July 1977, pp. 281–284.
Neubuser J., «Some Applications of Group Theoretical Programs»,Proceedings of the Second Symposium on Symbolic and Algebraic Manipulations, L. A., California, 1971,ACM, New York, 1971, p. 77.
Pearson E. S. andHartley H. O., eds., «Biometrika Tables for Statisticians», Vol. 1, Cambridge University Press, 1958, p. 122.
Pless V., «Encryption Schemes for Computer Confidentiality»,IEEE Trans. Comp., C-26, 11, November 1977, pp. 1133–1136.
Rabin M. O., «Probabilistic Algorithms»,Algorithms and Complexity, J. F. Straub (ed.), Academic Press, New York, 1976, pp. 21–40.
Shannon C. E., «The Mathematical Theory of Communication»,Bell Syst. J., 27, July and October 1948, pp. 379–423 and pp. 623–656.
Shannon C. E., «Communication Theory of Secrecy Systems»,Bell Syst. J., 28, October 1949, pp. 656–715.
Sims C. C., «Computational Methods in the Study of Permutation Groups», «Computational Problems in Abstract Algebra»,Proc. Conf., Oxford, 1964, Pergamon Press, Oxford, 1970, pp. 169–183.
Surkan A. J. andKlopping J., «Comparative Tests for RPGM», unpublished working draft.
Wielandt H., «Finite Permutation Groups», Academic Press, 1964.
Author information
Authors and Affiliations
Additional information
(Conferenza tenuta il 10 dicembre 1984) dal Prof. Magliveras
Rights and permissions
About this article
Cite this article
Magliveras, S.S., Oberg, B.A. & Surkan, A.J. A new random number generator from permutation groups. Seminario Mat. e. Fis. di Milano 54, 203–223 (1985). https://doi.org/10.1007/BF02924858
Issue Date:
DOI: https://doi.org/10.1007/BF02924858