An efficient square root computation in finite fields GF(p2d)

Feng Wang, Yasuyuki Nogami, Yoshitaka Morikawa

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


This paper focuses on developing a square root (SQRT) algorithm in finite fields GF(p2d)(d ≥ 0). Examining the Smart algorithm, a well-known SQRT algorithm, we can see that there is some computation overlap between the Smart algorithm and the quadratic residue (QR) test, which must be implemented before a SQRT computation. It makes the Smart algorithm inefficient. In this paper, we propose a new QR test and a new SQRT algorithm in GF(p2d), in which not only there is no computation overlap, but also most of computations required for the proposed SQRT algorithm in GF(p2d) can be implemented in the corresponding subfields GF(p2d-l) for 1 ≤ i ≤ d, which yields many reductions in the computational time and complexity. The computer simulation also shows that the proposed SQRT algorithm is much faster than the Smart algorithm.

Original languageEnglish
Pages (from-to)2792-2799
Number of pages8
JournalIEICE Transactions on Fundamentals of Electronics, Communications and Computer Sciences
Issue number10
Publication statusPublished - Oct 2005


  • Finite fields
  • Quadratic residue
  • Square root

ASJC Scopus subject areas

  • Signal Processing
  • Computer Graphics and Computer-Aided Design
  • Electrical and Electronic Engineering
  • Applied Mathematics


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