More Comprehensive Computational Number Theory

Computational Number Theory

Historically, computation has been a driving force in the development of mathematics. To help measure the sizes of their fields, the Egyptians invented geometry. To help predict the positions of the planets, the Greeks invented trigonometry. Algebra was invented to deal with equations that arose when mathematics was used to model the world. The list goes on, and it is not just historical. If an...

Math 788M: Computational Number Theory

Computational Number Theory and Algebra

Proof (Continued from the previous class) We have already shown in the previous class that the algorithm halts after polynomially many executions of the while-loop as the index i can decrease for at most O(n logA) times in step 6. In order to complete the proof, we also need to show that the size of the numerator and denominator of any rational number involved in the computation is polynomially...

Computational Number Theory and Applications to Cryptography

• Greatest common divisor (GCD) algorithms. We begin with Euclid’s algorithm, and the extended Euclidean algorithm [2, 12]. We will then discuss variations and improvements such as Lehmer’s algorithm [14], the binary algorithms [12], generalized binary algorithms [20], and FFT-based methods. We will also discuss how to adapt GCD algorithms to compute modular inverses and to compute the Jacobi a...

Topics in Computational Algebraic Number Theory

We describe practical algorithms from computational algebraic number theory, with applications to class field theory. These include basic arithmetic, approximation and uniformizers, discrete logarithms and computation of class fields. All algorithms have been implemented in the Pari/Gp system.