Polynomial ring transforms for efficient XOR-based erasure coding

Abstract

Abstract-The complexity of software implementations of MDS erasure codes mainly depends on the efficiency of the finite field operations implementation. In this paper, we propose a method to reduce the complexity of the finite field multiplication by using simple transforms between a field and a ring to perform the multiplication in a ring. We show that moving to a ring reduces the complexity of the operations. Then, we show that this construction allows the use of simple scheduling to reduce the number of operations.

Simplifying Forwarding Data Plane Operations with XOR-Based Source Routing

“…So, the number of operations in the field is s P /w × s i /w. Many efficient methods were proposed to optimize multiplications in finite fields (see e.g., [23], [24]), but it can generally be assumed that a multiplication in F 2 w can be done in O(w 2 ) bit operations. It follows that the complexity of the filtering operation is in O(s P .s i ) number of bit operations and, thus, does not depend on w.…”