Recursive Double-Size Fixed Precision Arithmetic

We propose a new fixed precision arithmetic package called RecInt. It uses a recursive double-size data-structure. Contrary to arbitrary precision packages like GMP, that create vectors of words on the heap, RecInt large integers are created on the stack. The space allocated for these integers is a power of two and arithmetic is performed modulo that power. Operations are thus easily implemented recursively by a divide and conquer strategy. Among those, we show that this packages is particularly well adapted to Newton-Raphson like iterations or Montgomery reduction. Recursivity is implemented via doubling algorithms on templated data-types. The idea is to extend machine word functionality to any power of two and to use template partial specialization to adapt the implemented routines to some specific sizes and thresholds. The main target precision is for cryptographic sizes, that is up to several tens of machine words. Preliminary experiments show that good performance can be attained when comparing to the state of art GMP library: it can be several order of magnitude faster when used with very few machine words. This package is now integrated within the Givaro C++ library and has been used for efficient exact linear algebra computations.