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$$a \cdot 2^e + b \equiv a \cdot c + b \pmod{2^e-c}$$  If the binary representation of $c$ is simple enough, we can compute $a \cdot c$ without doing a full multiply, and hence we can compute this modular reduction significantly faster than we could for an arbitrary prime. This allows us to compute the modular multiplication of two numbers in not much more time than it takes to perform a bignum multiplication of those two numbers. Examples of Elliptic Curves that allow this speed up include the so-called NIST curves\cite{NIST_1999}, Curve25519\cite{Bern_2005}, and the Microsoft NUMS curves\cite{Black_2014}.  If we instead select an aribtrary prime without such a simple representation, there are still some optimizations we can do beyond the obvious 'perform a multiply, and then perform a generic modulo reduction'. We can implement Montgomery Multiplication, which replaces the modulo operation with some multiplies and shifts; the net result is that a modular reduction can be done in the time of approximately two bignum multiplications. multiplications; in other words, modular reduction of a special form prime can be done approximately twice as fast as an arbitrary prime.  - Side channel attacks  - Scalar randomizatoin