EUROCRYPT 1982: Cryptography pp 71-128

Encrypting by Random Rotations

  • N. J. A. Sloane
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 149)

Abstract

This paper gives some well-known, little known, and new results on the problem of generating random elements in groups, with particular emphasis on applications to cryptography. The groups of greatest interest are the group of all orthogonal n × n matrices and the group of all permutations of a set. The chief application is to A. D. Wyner’s analog scrambling scheme for voice signals.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Abramowitz, M. and Stegun, I. A. (1972), Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, US. Dept. Commerce, Washington, D.C.MATHGoogle Scholar
  2. Ahrens, J. H. and Dieter, U. (1972), Computer methods for sampling from the exponential and normal distributions, Commun. ACM, 15, 873–882.MATHCrossRefMathSciNetGoogle Scholar
  3. Ahrens, J. H. and Dieter, U. (1973). Extensions of Forsythe’s method for random sampling from the normal distribution, Math. Comp., 27, 927–937.MATHCrossRefMathSciNetGoogle Scholar
  4. Atkinson, A. C. and Pearce, M. C. (1976), The computer generation of beta, gamma and normal random variables, J. Royal Statist. Soc., A139, 431–461.MathSciNetGoogle Scholar
  5. Auerbach, H. (1933–34), Sur les groups linéaires bornés, Studia Math., 4, 113–127 and 158–166; 5, 43–49.MATHGoogle Scholar
  6. Ayoub, F. (1981), Encryption with keyed random permutations, Electronics Letters, 17, 583–585.CrossRefMathSciNetGoogle Scholar
  7. Baer, R. M. and Brock, P. (1968), Natural sorting over permutation spaces, Math. Comp., 22, 385–410.CrossRefMathSciNetGoogle Scholar
  8. Berlekamp, E. R. (1968), Algebraic Coding Theory, McGraw-Hill, New York.MATHGoogle Scholar
  9. Bernhard, R. (1982), Breaching system security, IEEE Spectrum, 19 (No. 6), 24–31.Google Scholar
  10. Bloomfield, P. (1976), Fourier Analysis of Time Series: An Introduction. Wiley, New York.MATHGoogle Scholar
  11. Blum, L., Blum, M. and Shub, M. (1982), A simple secure pseudo-random number generator, presented at “Crypto 82”, Univ. of Calif., Santa Barbara, August 1982.Google Scholar
  12. Boothby, W. M. and Weiss, G. L., editors (1972), Symmetric Spaces, Dekker, New York.MATHGoogle Scholar
  13. Bourbaki, N. (1968), Groups et algebras de Lie, Chap. 4–6, Hermann, Paris.Google Scholar
  14. Bovey, J. D. (1980), The probability that some power of a permutation has small degree. Bull. London Math. Soc., 12, 47–51.MATHCrossRefMathSciNetGoogle Scholar
  15. Bovey, J. D. and Williamson, A. (1978), The probability of generating the symmetric group, Bull. London Math. Soc., 10, 91–96.MATHCrossRefMathSciNetGoogle Scholar
  16. Box, G. E. P. and Muller, M. E. (1958), A note on the generation of normal deviates, Annals Math. Stat., 29, 610–611.MATHCrossRefGoogle Scholar
  17. Brent, R. P. (1974), A Gaussian pseudo-random number generator, Commun. ACM, 17, 704–706.MATHCrossRefGoogle Scholar
  18. Brigham, E. O. (1974), The Fast Fourier Transform. Prentice-Hall, Englewood Cliffs, N.J.MATHGoogle Scholar
  19. Bright, H. S. and Enison, R. L. (1979), Quasi-random number sequences from a long-period TLP generator with remarks on application to cryptography, Computing Surveys, 11, 357–370.MATHCrossRefGoogle Scholar
  20. Brillinger, D. R. (1973), Time Series: Data Analysis and Theoty, Holt, Rinehart and Winston, New York.Google Scholar
  21. Brown, G. W. (1956), Monte Carlo methods, in Modern Mathematics for the Engineer, edited E. F. Beckenbach, McGraw-Hill, New York, pp. 279–303.Google Scholar
  22. Brown, M. and Solomon, H. (1979): On combining pseudorandom number generators, Ann. Statistics, 7, 691–695.MATHMathSciNetCrossRefGoogle Scholar
  23. Cartan, E. (1966), The Theory of Spinors, Hermann, Paris. Reprinted by Dover Publications, New York, 1981.MATHGoogle Scholar
  24. Chambers, R. P. (1967), Random-number generation, IEEE Spectrum, 4 (No. 2), 48–56.CrossRefGoogle Scholar
  25. Chatfield, C. (1975), The Analysis of Time Series: Theory and Practice, Chapman and Hall, London.MATHGoogle Scholar
  26. Conway, J. H., Parker, R. A. and Sloane, N. J. A. (1982), The covering radius of the Leech lattice, Proe. Royal Soc. London, A 380, 261–290.MATHMathSciNetCrossRefGoogle Scholar
  27. Cook, J. M. (1957). Rational formulae for the production of a spherically symmetric probability distribution, Math. Tables Other Aids Comp., 11, 81–82.MATHGoogle Scholar
  28. Cook, J. M. (1959), Remarks on a recent paper, Commun. ACM, 2 (No. 10), 26.Google Scholar
  29. Coppersmith, D. and Grossman, E. (1975), Generators for certain alternating groups with applications to cryptography, SIAM J. Applied Math., 29, 624–627.CrossRefMATHMathSciNetGoogle Scholar
  30. Coxeter, H. S. M. (1973), Regular Polytopes, Dover, New York, third edition.Google Scholar
  31. Davis, R. M. (1978), The Data Encryption Standard in perspective. IEEE Communications Society Magazine, 16 (November), 5–9.CrossRefGoogle Scholar
  32. Deak, I. (1979), Comparison of methods for genercting uniformly distributed random points in and on a hypersphere, Problems of Control and Information Theory, 8, 105–113.MATHMathSciNetGoogle Scholar
  33. Diaconis, P. (1980), Average running time of the fast Fourier transform, J. Algorithms, 1, 187–208.MATHCrossRefMathSciNetGoogle Scholar
  34. Diaconis, P. (1982), Group Theory in Statistics, lecture notes, Harvard University.Google Scholar
  35. Diaconis, P. and Graham, R. L. (1977), Spearman’s footrule as a measure of disarray, J. Royal Stat. Soc., B 39, 262–268.MATHMathSciNetGoogle Scholar
  36. Diaconis, P., Graham, R. L., and Kantor, W. M. (1982), The mathematics of perfect shuffles, Advances in Applied Math., in press.Google Scholar
  37. Diaconis, P. and Shahshahani, M. (1981), Generating a random permutation with random transpositions, Z. Wahrscheinlichkeitstheorie, 57, 159–179.MATHCrossRefMathSciNetGoogle Scholar
  38. Diaconis, P. and Shahshahani, M. (1982), Factoring probabilities on compact groups, preprint.Google Scholar
  39. Dieter, U. and Ahrens, J. H. (1973), A combinatorial method for the generation of normally distributed random variables, Computing, 11, 137–146.MATHCrossRefMathSciNetGoogle Scholar
  40. Diffie, W. and Hellman, M. E. (1976), A critique of the proposed Data Encryption Standard, Commun. ACM, 19, 164–165.Google Scholar
  41. Diffie, W. and Hellman, M. E. (1977), Exhaustive analysis of the NBS Data Encryption Standard, Computer, 10, 74–84.CrossRefGoogle Scholar
  42. Dixon, J. D. (1969), The probability of generating the symmetric group, Math. Zeit., 110, 199–205.MATHCrossRefMathSciNetGoogle Scholar
  43. Durstenfeld, R. (1964), Random permutation, Commun. ACM, 7, 420.CrossRefGoogle Scholar
  44. Eaton, M. L. and Perlman, M. (1977), Generating O(n) with reflections, Pacific J. Math., 73, 73–80.MathSciNetGoogle Scholar
  45. Even, S. and Goldreich, O. (1981), The minimum-length generator sequence problem is NP-hard, J. Algorithms, 2, 311–313.MATHCrossRefMathSciNetGoogle Scholar
  46. Feistel, H. (1973), Cryptography and computer privacy, Scientific American, 228 (May), 15–23.CrossRefGoogle Scholar
  47. Feistel, H., Notz, W. A. and Smith, J. L. (1975), Some cryptographic techniques for machine-to-machine data communications, Proc. IEEE, 63, 1545–1554.CrossRefGoogle Scholar
  48. Feller, W. (1957), An Introduction to Probability Theory and Its Applications, Volume I, Wiley, New York, second edition.MATHGoogle Scholar
  49. Fienberg, S. E. (1971), Randomization and social affairs: the 1970 draft lottery, Science, 167 (22 January), 255–261.CrossRefGoogle Scholar
  50. Fino, B. J. and Algazi, V. R. (1976), Unified matrix treatment of the fast Walsh-Hadamard transform, IEEE Trans. Computers, C-25, 1142–1146.MathSciNetCrossRefGoogle Scholar
  51. Fox, P. A., editor (1976), The PORT Mathematical Subroutine Library, Bell Laboratories, Murray Hill, New Jersey.Google Scholar
  52. Furstenberg, H. (1980), Random walks on Lie groups, in Harmonic Analysis and Representations of Semisimple Lie Groups, edited by J. A. Wolf et al., Reidel Publ., Dordrecht, Holland, pp. 467–489.Google Scholar
  53. Geffe, P. R. (1967), An open letter to communication engineers, Proc. IEEE, 55, 2173.CrossRefGoogle Scholar
  54. Geramita, A. V. and Seberry, J. (1979), Orthogonal Designs, Dekker, New York.MATHGoogle Scholar
  55. Girsdansky, M. B. (1971), Data privacy—cryptology and the computer at IBM Research, IBM Research Reports, 7 (No. 4), 12 pages. Reprinted in Computers and Automation, 21 (April, 1971), 12–19.Google Scholar
  56. Golomb, S. W. (1964), Random permutations, Bull. Amer. Math. Soc., 70, 747.MathSciNetCrossRefGoogle Scholar
  57. Goncharov, V. (1944), Du domaine d’analyse combinatoire (Russian, French summary), Bull. de 1’Académie URSS, Sér. Math. 8, 3–48. English translation in Amer. Math. Soc. Translations, (2) 19 (1962), 1–46.MATHGoogle Scholar
  58. Good, I. J. (1 958), The interaction algorithm and practical Fourier analysis, J. Roy. Stat. Soc. B 20, 361–372 and B 22, 372–375.MathSciNetGoogle Scholar
  59. Grenander, U. (1963), Probability on Algebraic Structures. Wiley, New York.Google Scholar
  60. Guivarc’h, Y., Keane, M. and Roynette, B. (1977), Marches aleatoires sur les groups de Lie, Lecture Notes in Math. 624, Springer-Verlag, New York.Google Scholar
  61. Hall, M., Jr. (1967), Combinatorial Theory, Blaisdell, Waltham, Mass.MATHGoogle Scholar
  62. Hall, M., Jr. (1975), Semi-automorphisms of Hadamard matrices, Math. Proc. Camb. Phil. Soc., 77, 459–473.MATHCrossRefGoogle Scholar
  63. Halmos, P. R. (1950), Measure Theory, Van Nostrand, Princeton, N.J.MATHGoogle Scholar
  64. Halmos, P. R. (1956), Lectures on Ergodic Theory, Chelsea, New York.MATHGoogle Scholar
  65. Hammersley, J. H. (1972), A few seedlings of research, in Proc. Sixth Berkeley Symp. Math. Stat. and Prob., Vol. 1, pp. 345–394.MathSciNetGoogle Scholar
  66. Hannan, E. J. (1960), Time Series Analysis, Methuen, London.MATHGoogle Scholar
  67. Harwit, M. and Sloane, N. J. A. (1979), Hadamard Transform Optics, Academic Press, New York.MATHGoogle Scholar
  68. Heiberger, R. M. (1978), Generation of random orthogonal matrices, Applied Statistics, 27, 199–206.MATHCrossRefGoogle Scholar
  69. Hess, P. and Wirl, K. (1983), A voice scrambling system for testing and demonstration, in this volume.Google Scholar
  70. Hewitt, E. and Ross, K. A. (1963–1970), Abstract Harmonic Analysis, 2 vols., Springer-Verlag, New York.MATHGoogle Scholar
  71. Heyer, H. (1977), Probablity Measures on Locally Compact Groups, Springer-Verlag, New York.Google Scholar
  72. Heyer, H., editor (1982), Probability Measures on Groups. Lecture Notes in Math. 928, Springer-Verlag, New York.MATHGoogle Scholar
  73. Hicks, J. S. and Wheeling, R. F. (1959), An efficient method for generating uniformly distributed points on the surface of an n-dimensional sphere, Commun. ACM, 2 (No.4), 17–19.MATHCrossRefGoogle Scholar
  74. Hopf, E. (1937), Ergodenthorie, J. Springer, Berlin. Reprinted by Chelsea, New York. 1948.Google Scholar
  75. Humphreys, J. E. (1972), Introduction to Lie Algebras and Representation Theory, Springer-Verlag, New York, second printing.MATHGoogle Scholar
  76. Ito, N., Leon, J. S. and Longyear, J. Q. (1981), Classification of 3-(24,12.5) designs and 24-dimensional Hadamard matrics, J. Combinatorial Theory, A31, 66–93.CrossRefMathSciNetGoogle Scholar
  77. Jansson, B. (1966), Random Number Generators, Stockholm.Google Scholar
  78. Jayant, N. S. (1982), Analog scramblers for speech privacy. preprint.Google Scholar
  79. Kantor, W. M. (1969), Automorphism groups of Hadamard matrices, J. Combinatorial Theory, 6, 279–281.MATHMathSciNetCrossRefGoogle Scholar
  80. Kantor, W. M. (1982), Polynomial-time perfect shuffling. preprint.Google Scholar
  81. Kendall, M. (1970), Rank Correlation Methods, Griffin, London, fourth edition.Google Scholar
  82. Kennedy, W. J., Jr. and Gentle, J. E. (1980), Statistical Computing, Dekker, New York.MATHGoogle Scholar
  83. Knop, R. E. (1970), Random vectors uniform in solid angle, Commun. ACM, 13, 326.CrossRefGoogle Scholar
  84. Knuth, D. E. (1980), Deciphering a linear congruential encryption, Report STAN-CS-80-800, Computer Science Dept., Stanford Univ., Stanford, Calif.Google Scholar
  85. Knuth, D. E. (1981), The Art of Computer Programming, Volume 2: Seminumerical Algorithms, Addison-Wesley, Reading Mass., second edition.Google Scholar
  86. Lewis, T. G. (1975), Distribution Sampling for Computer Simulation, Lexington Books, Lexington, Mass.MATHGoogle Scholar
  87. Li, T. Y. and Yorke, J. A. (1978), Ergodic maps on [0,1] and nonlinear pseudorandom number generators, Nonlinear Analysis, Theory, Methods and Applications, 2, 473–481.MATHCrossRefMathSciNetGoogle Scholar
  88. Lloyd, S. P. (1977), Random rotation secrecy systems, unpublished memorandum, Bell Laboratories, Murray Hill, N.J.Google Scholar
  89. Lloyd, S. P. (1978), Choosing a rotation at random, unpublished memorandum, Bell Laboratories, Murray Hill, N.J.Google Scholar
  90. Logan, B. F. and Shepp, L. A. (1977), A variational problem for random Young tableaux, Advances in Math., 26, 206–222.MATHCrossRefMathSciNetGoogle Scholar
  91. McGonegal, C. A., Berkley, D. A. and Jayant, N. S. (1981), Private communications, Bell Syst. Tech. J. 60, 1563–1572.Google Scholar
  92. MacKinnon, N. R. F. (1980), The development of speech encipherment, Radio and Electronic Engineer, 50, No. 4, 147–155.CrossRefGoogle Scholar
  93. MacLaren, M. D. and Marsaglia (1965), Uniform random number generators, J. Assoc. Comput. Mach., 12, 83–89.MATHMathSciNetGoogle Scholar
  94. MacWilliams, F. J. and Sloane, N. J. A. (1981), The Theory of Error-Correcting Codes, North-Holland, Amsterdam.Google Scholar
  95. Marsaglia, G. (1972), Choosing a point from the surface of a sphere, Annals. Math. Stat., 43, 645–646.MATHCrossRefGoogle Scholar
  96. Marsaglia, G., Ananthanarayanan, K., and Paul, N. (1973), Random number generator package — “Super-Duper”, School of Computer Science, McGill University, Montreal, Quebec.Google Scholar
  97. Marsaglia, G., Ananthanarayanan, K., and Paul, N. J. (1976), Improvements on fast methods for generating normal random variables, Information Processing Letters, 5 (No. 2), 27–30.MATHCrossRefMathSciNetGoogle Scholar
  98. Marsaglia, G. and Bray, T. A. (1964), A convenient method for generating normal variables, SLAM Review, 6, 260–264.MATHCrossRefMathSciNetGoogle Scholar
  99. Massey, J. L. (1969), Shift-register synthesis and BCH decoding, IEEE Trans. Inform. Theory, IT-15, 122–127.CrossRefMathSciNetGoogle Scholar
  100. Meyer, C. H. and Tuchman, W. L. (1972), Pseudorandom codes can be cracked, Electronic Design, 20 (Nov. 9), 74–76.Google Scholar
  101. Mihram, G. A. (1972), Simulation: Statistical Foundations and Methodology, Academic Press, New York.MATHGoogle Scholar
  102. Moore, C. C., editor (1973), Harmonic Analysis on Homogeneous Spaces, Proc. Sympos. Pure Math. 26, Amer. Math, Soc., Providence, Rhode Island.Google Scholar
  103. Morris, R. (1978), The Data Encryption Standard — retrospective and prospects, IEEE Communications Society Magazine, 16 (November), 11–14.CrossRefGoogle Scholar
  104. Morris, R., Sloane, N. J. A. and Wyner, A. D. (1977), Assessment of the National Bureau of Standards proposed Federal Data Encryption Standard, Cryptologia, 1, 281–306.CrossRefGoogle Scholar
  105. Muller, M. E. (1959), A note on a method for generating points uniformly on n-dimensional spheres, Commun. ACM, 2 (No. 4), 19–20.MATHCrossRefGoogle Scholar
  106. Von Neumann, J. (1951), Various techniques used in connection with random digits, in Monte Carlo Methods, National Bureau of Standards Applied Math. Series 12, U. S. Dept. Commerce, Washington, D.C. pp. 36–38.Google Scholar
  107. Niederreiter, H. (1978), Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc., 84, 957–1041.MATHMathSciNetCrossRefGoogle Scholar
  108. Nijenhuis, A. and Wilf, H. S. (1978), Combinatorial Algorithms, Academic Press, New York, second edition.MATHGoogle Scholar
  109. Page, E. S. (1967), A note on generating random permutations, Applied Statist., 16, 273–274.CrossRefGoogle Scholar
  110. Plackett, R. L. (1968), Random permutations. J. Royal Stat. Soc., 30, 517–534.MATHMathSciNetGoogle Scholar
  111. Pratt, W. K. (1969), An algorithm for a fast Hadamard matrix transform of order twelve, IEEE Trans. Computers, C-18, 1131–1132.CrossRefGoogle Scholar
  112. Pratt, W. K., Kane, J. and Andrews, H. C. (1969), Hadamard transform image coding, Proc. IEEE, 57, 58–68.CrossRefGoogle Scholar
  113. Reeds, J. (1977), “Cracking” a random number generator, Cryptologia, 1 (No. 1), 20–26.CrossRefMathSciNetGoogle Scholar
  114. Reeds, J. (1979), Cracking a multiplicative congruential encryption algorithm. in Information Linkage Between Applied Mathematics and Industry (Proc. First Annual Workshop, Naval Postgraduate School, Monterey, Calif., 1978), Academic Press, New York, pp. 467–472.Google Scholar
  115. Reeds, J. (1979a), Solution of challenge cipher, Cryptologia, 3, 83–95.CrossRefGoogle Scholar
  116. Riordan, J. (1958), An Introduction to Combinatorial Analysis, Wiley, New York.MATHGoogle Scholar
  117. Robbins, D. P. and Bolker, E. D. (1981), The bias of three pseudo-random shuffles. Aequationes Math., 22, 268–292.MATHCrossRefMathSciNetGoogle Scholar
  118. Rose, D. J. (1980), Matrix identities of the fast Fourier transform, Linear Alg. Applic., 29, 423–443.MATHCrossRefGoogle Scholar
  119. Rosenblatt, J. R. and Filliben, J. J. (1971), Randomization and the draft lottery, Science, 167 (22 January), 306–308.CrossRefGoogle Scholar
  120. Sakasegawa, H. (1978), On generation of normal pseudo-random numbers. Ann. Inst. Statist. Math., A30, 271–279.CrossRefMathSciNetGoogle Scholar
  121. Schmeiser, B. W. (1980), Random variate generation: a survey, in Simulation with Discrete Models: A State-of-the-Art Survey, edited by T. I. Oren, C. M. Shub and P. F. Roth, IEEE Press, New York.Google Scholar
  122. Schrack, G. F. (1972), Remark on Algorithm 381, Commun. ACM, 15, 468.CrossRefGoogle Scholar
  123. Schwerdtfeger, H. (1950), Introduction to Linear Algebra and the Theory of Matrices, Noordhoff, Groningen.MATHGoogle Scholar
  124. Shamir, A. (1981), The generation of cryptographically strong pseudo-random sequences, presented at “Crypto 81”, Univ. of Calif., Santa Barbara, August 1981.Google Scholar
  125. Shannon, C. E. (1949), Communication theory of secrecy systems, Bell Syst. Tech. J., 28, 656–715.MathSciNetMATHGoogle Scholar
  126. Shepp, L. A. and Lloyd, S. P. (1966), Ordered cycle lengths in a random permutation, Trans. Amer. Math. Soc., 121, 340–357.MATHCrossRefMathSciNetGoogle Scholar
  127. Sibuya, M. (1964), A method for generating uniformly distributed points on n-dimensional spheres, Ann. Inst. Stat. Math., 14, 81–85.MathSciNetCrossRefGoogle Scholar
  128. Slepian, D. (1978), Prolate spheroidal wave functions. Fourier analysis, and uncertainty, Part V: the discrete case, Bell Syst. Tech. J., 57, 1371–1430.Google Scholar
  129. Sloane, N. J. A. (1981), Error-correcting codes and cryptography, in The Mathematical Gardner, edited by D. A. Klarner, Prindle, Weber and Schmidt, Boston, pp. 346–382. Reprinted in Cryptologia, 6 (1982), 128–153 and 258–278.Google Scholar
  130. Smith, J. L. (1971), The design of Lucifer. a cryptographic device for data communicazions, Report RC-3326, IBM Thomas Watson Research Center, Yorktown Heights, N.Y.Google Scholar
  131. Stewart, G. W. (1980), The efficient generation of random orthogonal matrices with an application to condition estimators, SIAM J. Numer. Anal., 17, 403–409.MATHCrossRefMathSciNetGoogle Scholar
  132. Tashiro, Y. (1977), On methods for generating uniform random points on the surface of a sphere. Ann. Inst. Stat. Math., A29, 295–300.CrossRefMathSciNetGoogle Scholar
  133. Turyn, R. J. (1974), Hadamard matrices, Baumert-Hall units, four-symbol sequences, pulse compression, and surface wave encodings, J. Combinatorial Theory, 16A, 313–333.CrossRefMathSciNetGoogle Scholar
  134. Veršik, A. M. and Kerov, S. V. (1977), Asymptotics of the Plancherel measure of the symmetric group and the limiting form of Young tableaux (Russian), Dokl. Akad. Nauk SSSR, 233, No. 6. English translation in Soviet Math. Doklady, 18 (1977), 527–531.Google Scholar
  135. Wallis, W. D., Street, A. P. and Wallis, J. S. (1972), Combinatorics: Room Squares, Sum-Free Sets, Hadamard Matrices, Lecture Notes in Math., 292, Springer-Verlag, New York.Google Scholar
  136. Walter, P. (1981), An Introduction to Ergodic Theory, Springer-Verlag, Berlin and New York.Google Scholar
  137. Warner, G. (1972), Harmonic Analysis on Semi-Simple Lie Groups, 2 vols., Springer-Verlag, New York.Google Scholar
  138. Wyner, A. D. (1979), An analog scrambling scheme which does not expand bandwidth, Part I: discrete time, IEEE Trans. Inform. Theory, IT-25, 261–274.CrossRefMathSciNetGoogle Scholar
  139. Wyner, A. D. (1979a), An analog scrambling scheme which does not expand bandwidth. Part II: continuous times, IEEE Trans. Inform. Theory, IT-25, 415–425.CrossRefMathSciNetGoogle Scholar
  140. Yao, A. C. (1982), private communication.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1983

Authors and Affiliations

  • N. J. A. Sloane
    • 1
  1. 1.Bell LaboratoriesMathematics and Statistics Research CenterMurray HillUSA

Personalised recommendations