Abstract
Abstract This paper presents an efficient algorithm for computing 11th-power residue symbols in the cyclo-tomic field ℚ ( ζ 11 ) , $ \mathbb{Q}\left( {{\zeta }_{11}} \right), $ where 11 is a primitive 11th root of unity. It extends an earlier algorithm due to Caranay and Scheidler (Int. J. Number Theory, 2010) for the 7th-power residue symbol. The new algorithm finds applications in the implementation of certain cryptographic schemes.
Highlights
Quadratic and higher-order residuosity is a useful tool that finds applications in several cryptographic constructions
For the case p = 2, it is well known that the Jacobi symbol can be computed by combining Euclid’s algorithm with quadratic reciprocity and the complementary laws for −1 and 2; see e.g. [10, Chapter 1]
The computation of the Jacobi symbol n proceeds by repeatedly performing 3 steps: (i) reduce a modulo n so that the result is smaller than n/2, (ii) extract the sign and the powers of 2 for which the symbol is calculated explicitly with the complementary laws, and (iii) apply the reciprocity law resulting in the ‘numerator’ and ‘denominator’ of the symbol being flipped
Summary
Quadratic and higher-order residuosity is a useful tool that finds applications in several cryptographic constructions. Caranay and Scheidler describe a generic algorithm in [3, Section 7] for computing the pth-power residue symbol for any prime p ≤ 11, building on Lenstra’s norm-Euclidean algorithm. They provide a detailed implementation for the case p = 7. The contributions of this paper are three-fold: We provide explicit conditions for primary algebraic integers in Z[ζ11]; we devise an efficient algorithm for finding a primary associate; and we give explicit complementary laws for a set of four fundamental units and for the special prime 1 − ζ11. We present the ingredients and develop the companion algorithms for the computation of the eleventh power residue symbol
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