Diffie-Hellman Primitives in SP800-56A I wonder if someone can give an explain about the different between two Diffie-Hellman Primitives defined in SP800-56A, CH5.7.1

5.7.1.1 Finite Field Cryptography Diffie-Hellman (FFC DH) Primitive
5.7.1.2 Elliptic Curve Cryptography Cofactor Diffie-Hellman (ECC CDH) Primitive

Actually the first one is named as (FFC DH) and its looked very similar to RSA way.
And for the 2nd one , though it is based on ECC, but is also on a Finite Field.
A group if a set of elements and some operation that satisfy some requirements. This operation is usually called "addition" or "multiplication", depending on the group, even though it may not be actually the classic addition or multiplication. The integers $0 &lt; x &lt; p$ for some prime $p$ form a group under the modular multiplication operation (multiply the integers, then compute the remainder of the division by $p$). This group can be denoted by $\mathbb{Z}_p^*$.

A field is a set of elements with two operations that satisfy some requirements; you can think about it as consisting of two groups (one for each operation). The integers $0 &lt;= x &lt; p$ for some prime $p$ form a field under modular addition and multiplication. This field can be denoted by $\mathbb{F}_p$.

That is the field that is used by FFC DH. However, note that DH only requires a group, and not a field. FFC DH uses the multiplicative group of the finite field, i.e. $\mathbb{Z}_p^*$; the addition operation is never used.

An elliptic curve is a set of $(x, y)$ coordinates ("points") that satisfy some equation. In cryptography, these coordinates are elements of a finite field, e.g. $\mathbb{F}_p$. The curve also forms a group under the "point addition" operation, which takes two pair of points and return a third point also on the curve. Note that a curve forms a group, and uses an underlying field, but it's not a field itself.

(A curve has a "point multiplication" operation, but that's not a group operation between points, it's simply repeated addition, like exponentiation in $\mathbb{F}_p$ is just repeated multiplication).