Quasi-linear MBB evaluation of known sparse polynomials at large modulus and degree

Develop a quasi-linear time algorithm for modular black box (MBB) evaluation of a known sparse integer polynomial when both the polynomial’s degree and the evaluation modulus are large; specifically, compute f(a) mod m with total bit complexity quasi-linear in the bit-length of f and log m under these conditions.

Background

The paper relies on a multi-point modular derivative black box (MDBB) model to enable efficient subroutines. When attempting to update an unknown polynomial black box with an explicitly computed partial result, the authors identify a limitation of the traditional MBB model: efficient evaluation of a known sparse polynomial at large modulus and degree is not known to be achievable in quasi-linear time.

This gap motivates the MDBB approach and is explicitly stated as an unresolved capability in the MBB framework.