# Finite Field Arithmetic _ Montgomery reduction

In an attempt to understand the mathematical operations related to encryption with elliptic curves, in particular finite field arithmetic (Modular reduction) I found in the Montgomery reduction that Montgomery form is an alternative finite field representation. what is this suppose to mean ? what it does differ from the usual finite field representation ?

Montgomery arithmetic is a technique for modular computation modulo some integer $$m$$. It centers on Montgomery reduction, an alternative to reduction modulo $$m$$ by Euclidean division.
Montgomery reduction uses an alternate representation of the ring of integers modulo $$m$$. That is, the representation of a ring element $$A$$ as digits is not the same as in the Positional Number System, where $$A$$ is represented by digits $$a_j$$ in some base $$\beta$$, with $$0\le a_j<\beta$$ and $$A=\sum a_j\,\beta^j$$ (and perhaps $$A, making the tuple unique). In Montgomery representation with $$m<\beta^\ell$$ and $$\gcd(m,\beta)=1$$, a ring element $$A$$ is represented by a tuple of $$\ell$$ digits $$a_j$$, with $$0\le a_j<\beta$$ and $$\beta^\ell\,A\;\equiv\;\sum a_j\,\beta^j\pmod m$$ (and perhaps $$\sum a_j\,\beta^j, making the tuple unique).
Montgomery reduction has little relation with Montgomery elliptic curves. The closest is that for Montgomery elliptic curves on a finite field of large characteristic (that is field $$\mathbb F_{p^k}$$ with large $$p$$), arithmetic modulo $$p$$ can use Montgomery reduction.