Cryptanalysis of the Dickson-scheme
Proc. of a workshop on the theory and application of cryptographic techniques on Advances in cryptology---EUROCRYPT '85
A course in number theory and cryptography
A course in number theory and cryptography
Prime numbers and computer methods for factorization (2nd ed.)
Prime numbers and computer methods for factorization (2nd ed.)
Factors of generalized Fermat numbers
Mathematics of Computation
Algorithmic number theory
A survey of fast exponentiation methods
Journal of Algorithms
Handbook of Applied Cryptography
Handbook of Applied Cryptography
Efficient Algorithms for Computing the Jacobi Symbol
ANTS-II Proceedings of the Second International Symposium on Algorithmic Number Theory
On Probable Prime Testing and the Computation of Square Roots mod n
ANTS-IV Proceedings of the 4th International Symposium on Algorithmic Number Theory
Note on taking square-roots modulo N
IEEE Transactions on Information Theory
Improved generalized Atkin algorithm for computing square roots in finite fields
Information Processing Letters
Efficient Finite Fields in the Maxima Computer Algebra System
WAIFI '08 Proceedings of the 2nd international workshop on Arithmetic of Finite Fields
Improved generalized Atkin algorithm for computing square roots in finite fields
Information Processing Letters
A family of implementation-friendly BN elliptic curves
Journal of Systems and Software
Adleman-Manders-Miller root extraction method revisited
Inscrypt'11 Proceedings of the 7th international conference on Information Security and Cryptology
A Complete Generalization of Atkin's Square Root Algorithm
Fundamenta Informaticae
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In this paper, two improvements for computing square roots in finite fields are presented. Firstly, we give a simple extension of a method by O. Atkin, which requires two exponentiations in FMq, when q≡9 mod 16. Our second method gives a major improvement to the Cipolla–Lehmer algorithm, which is both easier to implement and also much faster. While our method is independent of the power of 2 in q−1, its expected running time is equivalent to 1.33 as many multiplications as exponentiation via square and multiply. Several numerical examples are given that show the speed-up of the proposed methods, compared to the routines employed by Mathematica, Maple, respectively Magma.