### Bulk execution of Euclidean algorithms on the CUDA-enabled GPU

#### Abstract

The bulk execution of a sequential algorithm is to execute it for many different inputs in turn or at the same time. A sequential algorithm is oblivious if the address accessed at each time unit is independent of the input. It is known that the bulk execution of an oblivious sequential algorithm can be implemented to run on a GPU very efficiently. The main purpose of our work is to implement the bulk execution of a Euclidean algorithm computing the GCD (Greatest Common Divisor) of two large numbers in a GPU. We first present a new efficient Euclidean algorithm that we call the Approximate Euclidean algorithm. The idea of the Approximate Euclidean algorithm is to compute an approximation of quotient by just one 64-bit division and to use it for reducing the number of iterations of the Euclidean algorithm. Unfortunately, the Approximate Euclidean algorithm is not oblivious. To show that the bulk execution of the Approximate Euclidean algorithm can be implemented efficiently in the GPU, we introduce a semi-oblivious sequential algorithms, which is almost oblivious. We show that the Approximate Euclidean algorithm can be implemented as a semi-oblivious algorithm. The experimental results show that our parallel implementation of the Approximate Euclidean algorithm for 1024- bit integers running on GeForce GTX Titan X GPU is 90 times faster than the Intel Xeon CPU implementation.

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