Almost Primality of Group Orders of Elliptic Curves Defined over Small Finite Fields

@article{Koblitz2001AlmostPO,
  title={Almost Primality of Group Orders of Elliptic Curves Defined over Small Finite Fields},
  author={Neal Koblitz},
  journal={Experimental Mathematics},
  year={2001},
  volume={10},
  pages={553 - 558}
}
  • N. Koblitz
  • Published 1 January 2001
  • Mathematics
  • Experimental Mathematics
Let E be an elliptic curve defined over a small finite field Fq, and let p be a prime number. We give a conjectural formula for the probabil ity that the order of the quotient group E(Fqp )/E(Fq) is prime, and compare it with experimental data. The motivation for this study comes from public key cryptography. 
View on Taylor & Francis
projecteuclid.org
8 Citations
Background Citations
2
Methods Citations
2

8 Citations

Elliptic curves over finite fields : number theoretic and cryptographic aspects

TLDR
Several natural questions about elliptic curves are presented, mostly over finite fields, that have led to some interesting number theoretic questions and whose solutions require rather involved techniques from various area of number theory.

Collisions in Fast Generation of Ideal Classes and Points on Hyperelliptic and Elliptic Curves

  • T. LangeI. Shparlinski
  • Mathematics, Computer Science
    Applicable Algebra in Engineering, Communication and Computing
  • 2004
TLDR
An upper bound on the number of collisions for this alternative approach to Koblitz curves is given and these bounds are tight for cyclic subgroups of prime order, which is the case of most practical interest for cryptography.

Divisibility, Smoothness and Cryptographic Applications

TLDR
This paper presents various properties of smooth numbers relating to their enumeration, distribution and occurrence in various integer sequences, and turns its attention to cryptographic applications in which smooth numbers play a pivotal role.

Topics in Geometry, Coding Theory and Cryptography

TLDR
This book presents survey articles on some of the new developments in the theory of algebraic function fields over finite fields, which have not yet been presented in other books or survey articles.

Finite fields of low characteristic in elliptic curve cryptography

This is to certify that the thesis prepared By Entitled Complies with University regulations and meets the standards of the Graduate School for originality and quality Forthedegreeof Final examining…

Some applications of coding theory in cryptography

TLDR
The final author version and the galley proof are versions of the publication after peer review that features the final layout of the paper including the volume, issue and page numbers.

GROUP STRUCTURE OF ELLIPTIC CURVES OVER FINITE FIELDS AND APPLICATIONS

Good and Bad Uses of Elliptic Curves in Cryptography

TLDR
The construction of cryptosystems using elliptic curves is described, the Elliptic Curve Discrete Logarithm Problem is discussed, and the different types of elliptic curve that can be chosen for cryptographic applications are surveyed.

References

SHOWING 1-10 OF 11 REFERENCES

Elliptic curve cryptosystems

TLDR
The question of primitive points on an elliptic curve modulo p is discussed, and a theorem on nonsmoothness of the order of the cyclic subgroup generated by a global point is given.

Nonsingular plane cubic curves over finite fields

  • R. Schoof
  • Mathematics
    J. Comb. Theory, Ser. A
  • 1987

Efficient Arithmetic on Koblitz Curves

  • J. Solinas
  • Computer Science, Mathematics
    Des. Codes Cryptogr.
  • 2000
TLDR
An improved version of theoblitz algorithm, which runs 50 times faster than any previous version, is given, based on a new kind of representation of an integer, analogous to certain kinds of binary expansions.

The Number of Prime Divisors of Certain Mersenne Numbers

It is the purpose of this paper to describe two tests of a modified form of this conjecture. It is known that all divisors of Mp must be of the form 2kp + 1 and simultaneously of the form 8k' ? 1,…

Divisors of Mersenne numbers

We add to the heuristic and empirical evidence for a conjecture of Gillies about the distribution of the prime divisors of Mersenne numbers. We list some large prime divisors of Mersenne numbers Mp…

Algebraic aspects of cryptography

  • N. Koblitz
  • Mathematics, Computer Science
    Algorithms and computation in mathematics
  • 1998
TLDR
1. Cryptography.- 1. Early History.; 2. The Idea of Public Key Cryptography.; 3. The RSA Cryptosystem.; 4. Diffie-Hellman and the Digital Signature Algorithm; 5. Secret Sharing, Coin Flipping, and Time Spent on Homework.; 6. Passwords, Signatures, and Ciphers; 7. Randomized Algorithms and Complexity Classes.

An Introduction to the Theory of Numbers

This is the fifth edition of a work (first published in 1938) which has become the standard introduction to the subject. The book has grown out of lectures delivered by the authors at Oxford,…

An introduction to the theory of numbersDivisors of Mersenne Clarendon

  • Math. Comp
  • 1979

United States (koblitz@math.washington.edu) Received

  • United States (koblitz@math.washington.edu) Received
  • 2000

Number fields, Universi- REFERENCES text

  • Number fields, Universi- REFERENCES text
  • 1977