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The least witness of a composite number

  • R. Balasubramanian
  • S. V. Nagaraj
Public-Key Cryptography
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1396)

Abstract

We consider the problem of finding the least witness of a composite number. If n is a composite number then a number w for which n is not a strong pseudo-prime to the base w is called a witness for n. Let w(n) be the least witness for a composite n. Bach [7] assuming the Generalized Riemann Hypothesis (GRH) showed that w(n) < 2log2 n. In this paper we are interested in obtaining upper bounds for w(n) without assuming the GRH.

Burthe [15) showed that w(n) = O(n1/(8√e)+ε) for all composite numbers n which are not a product of three distinct prime factors. For the three prime factor case he was able to show that w(n) = O(n1/(6√e)+). We improve his result to show w(n) = O(n1/(8√e)+) for all composite numbers n except Carmichael numbers n = pqr for which v2(p − 1) = v2 (q − 1) = v2 (r − 1). For the special Carmichaels we use an argument due to Heath-Brown to get w(n) = O(n1/(6.568√e)+).

We conjecture w(n) = O(n1/(8√e)+) for every composite number n and look at open problems. It appears to be very difficult to settle our conjecture.

Keywords

Deterministic Algorithm Average Running Time Composite Number Generalize Riemann Hypothesis Deterministic Polynomial Time 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • R. Balasubramanian
    • 1
  • S. V. Nagaraj
    • 1
    • 2
  1. 1.The Institute of Mathematical SciencesMadrasIndia
  2. 2.IBM Tokyo Research LaboratoryKanagawakenJapan

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