Period of the power generator and small values of Carmichael's function

被引:0
|
作者
Friedlander, JB [1 ]
Pomerance, C
Shparlinski, IE
机构
[1] Univ Toronto, Dept Math, Toronto, ON M5S 3G3, Canada
[2] Bell Labs, Dept Fundamental Math, Murray Hill, NJ 07974 USA
[3] Macquarie Univ, Dept Comp, Sydney, NSW 2109, Australia
关键词
Carmichael's function; RSA generator; Blum-Blum-Shub generator;
D O I
暂无
中图分类号
O29 [应用数学];
学科分类号
070104 ;
摘要
Consider the pseudorandom number generator u(n) = u(n-1)(e) (mod m), 0 less than or equal to u(n) less than or equal to m - 1, n = 1, 2,..., where we are given the modulus m, the initial value u(0) = v and the exponent e. One case of particular interest is when the modulus m is of the form pl, where p,l are different primes of the same magnitude. It is known from work of the first and third authors that for moduli m = pl, if the period of the sequence (u(n)) exceeds M3/4+epsilon, then the sequence is uniformly distributed. We show rigorously that for almost all choices of p,l it is the case that for almost all choices of v, e, the period of the power generator exceeds (pl)(1-epsilon). And so, in this case, the power generator is uniformly distributed. We also give some other cryptographic applications, namely, to ruling-out the cycling attack on the RSA cryptosystem and to so-called time-release crypto. The principal tool is an estimate related to the Carmichael function lambda (m), the size of the largest cyclic subgroup of the multiplicative group of residues modulo m. In particular, we show that for any Delta greater than or equal to (log log N)(3), we have lambda (m) greater than or equal to N exp(-Delta) for all integers m with 1 < m <less than or equal to> m less than or equal to N, apart from at most N exp (- 0.69(Delta log Delta)(1/3)) exceptions.
引用
收藏
页码:1591 / 1605
页数:15
相关论文
共 50 条
  • [31] Small values of the Euler function and the Riemann hypothesis
    Nicolas, Jean-Louis
    ACTA ARITHMETICA, 2012, 155 (03) : 311 - 321
  • [32] On the dynamics of a small-scale wind power generator
    Dosaev, M. Z.
    Samsonov, V. A.
    Seliutski, Yu. D.
    DOKLADY PHYSICS, 2007, 52 (09) : 493 - 495
  • [33] On the dynamics of a small-scale wind power generator
    M. Z. Dosaev
    V. A. Samsonov
    Yu. D. Seliutski
    Doklady Physics, 2007, 52 : 493 - 495
  • [34] On the period of the continued fraction for values of the square root of power sums
    Scremin, Amedeo
    ACTA ARITHMETICA, 2006, 123 (04) : 297 - 312
  • [35] A Novel Maximum Power Point Tracking Algorithm for Small Wind Power Generator
    Liu, Yepeng
    Liu, Weiliang
    Zhai, Yongjie
    PROCEEDINGS OF THE 2016 4TH INTERNATIONAL CONFERENCE ON ADVANCED MATERIALS AND INFORMATION TECHNOLOGY PROCESSING (AMITP 2016), 2016, 60 : 104 - 109
  • [36] Square values of Euler's function
    Pollack, Paul
    Pomerance, Carl
    BULLETIN OF THE LONDON MATHEMATICAL SOCIETY, 2014, 46 : 403 - 414
  • [37] ON THE FREQUENCY OF MULTIPLE VALUES OF A MEROMORPHIC FUNCTION OF SMALL ORDER
    SHEA, DF
    MICHIGAN MATHEMATICAL JOURNAL, 1985, 32 (01) : 109 - 116
  • [38] Power generator driven by Maxwell's demon
    Chida, Kensaku
    Desai, Samarth
    Nishiguchi, Katsuhiko
    Fujiwara, Akira
    NATURE COMMUNICATIONS, 2017, 8
  • [39] Small values of the Riemann zeta function on the critical line
    Kalpokas, Justas
    Sarka, Paulius
    ACTA ARITHMETICA, 2015, 169 (03) : 201 - 220
  • [40] Power generator driven by Maxwell’s demon
    Kensaku Chida
    Samarth Desai
    Katsuhiko Nishiguchi
    Akira Fujiwara
    Nature Communications, 8