On Euclid's Algorithm and the Computation of Polynomial Greatest Common Divisors
Journal of the ACM (JACM)
The Calculation of Multivariate Polynomial Resultants
Journal of the ACM (JACM)
Note on a Lower Bound on the Linear Complexity of the Fast Fourier Transform
Journal of the ACM (JACM)
Polynomial evaluation via the division algorithm the fast Fourier transform revisited
STOC '72 Proceedings of the fourth annual ACM symposium on Theory of computing
Chinese remainder and interpolation algorithms
SYMSAC '71 Proceedings of the second ACM symposium on Symbolic and algebraic manipulation
Exact solution of linear equations
SYMSAC '71 Proceedings of the second ACM symposium on Symbolic and algebraic manipulation
Fast modular transforms via division
SWAT '72 Proceedings of the 13th Annual Symposium on Switching and Automata Theory (swat 1972)
On the number of multiplications for the evaluation of a polynomial and all its derivatives
SWAT '72 Proceedings of the 13th Annual Symposium on Switching and Automata Theory (swat 1972)
On decreasing the computing time for modular arithmetic
SWAT '71 Proceedings of the 12th Annual Symposium on Switching and Automata Theory (swat 1971)
Efficient accelero-summation of holonomic functions
Journal of Symbolic Computation
Polynomial evaluation and interpolation on special sets of points
Journal of Complexity - Festschrift for the 70th birthday of Arnold Schönhage
Newton's method and FFT trading
Journal of Symbolic Computation
On the bit-complexity of sparse polynomial and series multiplication
Journal of Symbolic Computation
McBits: fast constant-time code-based cryptography
CHES'13 Proceedings of the 15th international conference on Cryptographic Hardware and Embedded Systems
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It is shown that if division and multiplication in a Euclidean domain can be performed in O(N log^aN) steps, then the residues of an N precision element in the domain can be computed in O(N log^a^+^1N) steps. A special case of this result is that the residues of an N precision integer can be computed in O(N log^2N log log N) total operations. Using a polynomial division algorithm due to Strassen [24], it is shown that a polynomial of degree N-1 can be evaluated at N points in O(N log^2N) total operations or O(N log N) multiplications. Using the methods of Horowitz [10] and Heindel [9], it is shown that if division and multiplication in a Euclidean domain can be performed in O(N log^aN) steps, then the Chinese Remainder Algorithm (CRA) can be performed in O(N log^a^+^1N) steps. Special cases are: (a) the integer CRA can be performed in O(N log^2N log log N) total operations, and (b) a polynomial of degree N-1 can be interpolated in O(N log^2N) total operations or O(N log N) multiplications. Using these results, it is shown that a polynomial of degree N and all its derivatives can be evaluated at a point in O(N log^2N) total operations.