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Title
Abstract
Key Takeaways
Introduction
Existing Square Root Methods
Kong et AL., Method
The Proposed Method
References

An Efficient Method for Finding Square Root

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Abstract

In this paper, we look into some methods for finding square roots that need more than one exponentiation in finite field Fq . Our proposed method calculates the primitive th e 2  root so that e is a biggest positive integer, and is suitable for cases if e is small. The proposed method enhances a related exponentiation caused from a well revised exponent, because it needs one exponentiation for computing the square root and is competitive compared with other existing methods. It is the first development introduced to well-known methods regarding rapidity of an average and the decrease of complexity of an algorithm.

Key takeaways
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  1. The proposed method requires only one exponentiation to compute square roots in finite field Fq.
  2. This method significantly simplifies calculations for small positive integers e.
  3. Existing methods like Shanks and Lehmer have higher complexities of O(log4 q) and O(log3 q), respectively.
  4. The Atkin method can compute square roots with just one exponentiation under certain conditions.
  5. The proposed method is particularly efficient for small values of e, especially for cases where e equals 2, 3, or 4.

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References (9)

  1. D. Shanks,"Five Number-Theoretic Algorithms", Proceeding of Second Manitoba Conference of Numerical Mathematics, pp.51-70, 1972
  2. D. Lehmer, "Computer technology applied to the theory of numbers", In Studies in Number Theory, Prentice-Hall Enblewood Cliffs, NJ pp.117-151, 1969.
  3. D. Han, D. Choi, and H. Kim, "Improved Computation of Square roots in Specific Finite Fields", IEEE Transactions on Computers, vol. 58, no. 2, pp.188-196, 2009.
  4. N. Nishihara, R. Harasawa, Y. Sueyoshi, and A. Kudo, "A remark on the computation of cube roots in finite fields", preprint, available at http://eprint.iacr.org/2009/457.
  5. G.Adj, and F.Rodrıguez-Henrıquez,"Square root computation over even extension fields", preprint, available at http://eprint.iacr.org/2012/685
  6. P.S.L.M. Barreto and J.F. Voloch, "Efficient Computation of Roots in Finite Fields," J. Design, Codes and Cryptography, vol. 39, pp. 275-280, 2006
  7. A. Atkin, "Probabilistic primality testing", summary by F. Morain, Inria Research Report 1779, pp.159- 163, 1992.
  8. S. Muller, "On the Computation of Square Roots in Finite Fields", Designs, Codes and Cryptography, vol. 31, pp. 301-312, 2004
  9. F. Kong, Z. Cai, J. Yu, and D. Li, "Improved Generalized Atkin Algorithm for Computing Square Roots in Finite Fields", Information Processing Letters, vol. 98, no. 1, pp. 1-5, 2006.

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