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Mathematics of Computation

Published by the American Mathematical Society, the Mathematics of Computation (MCOM) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6842 (online) ISSN 0025-5718 (print)

The 2020 MCQ for Mathematics of Computation is 1.98.

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A refinement of H. C. Williams’ $q$th root algorithm
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by Kenneth S. Williams and Kenneth Hardy PDF
Math. Comp. 61 (1993), 475-483 Request permission


Let p and q be primes such that $p \equiv 1 \pmod q$. Let a be an integer such that ${a^{(p - 1)/q}} \equiv 1 \pmod p$. In 1972, H. C. Williams gave an algorithm which determines a solution of the congruence ${x^q} \equiv a \pmod p$ in $O({q^3}\log p)$ steps, once an integer b has been found such that ${({b^q} - a)^{(p - 1)/q}} \nequiv 0,1 \pmod p$. A step is an arithmetic operation $\pmod p$ or an arithmetic operation on q-bit integers. We present a refinement of this algorithm which determines a solution in $O({q^4}) + O({q^2}\log p)$ steps, once b has been determined. Thus the new algorithm is better when q is small compared with p.
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Additional Information
  • © Copyright 1993 American Mathematical Society
  • Journal: Math. Comp. 61 (1993), 475-483
  • MSC: Primary 11A15; Secondary 11A07, 11Y16
  • DOI:
  • MathSciNet review: 1182249