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+SPAKE2 In Golang: Finite fields of Elliptic Curve
+#################################################
+
+:date: 2018-07-29 19:02 +0530
+:slug: golang_spake2_3
+:tags: go, golang, spake2, cryptography, ecc
+:author: copyninja
+:status: draft
+:summary: Third post in SPAKE2 in Golang. This post is my notes on finite fields
+          in elliptic curve group.
+
+Finite Fields
+=============
+
+*Finite field* or also called *Galois Field* is a set with finite number of
+elements. An example we can give is *integer modulo `p`* where `p` is prime.
+Finite fields can be denoted as :math:`\mathbb Z/p, GF(p)` or :math:`\mathbb
+F_p`.
+
+Finite fields will have 2 operations addition and multiplications. These
+operations are closed, associative and commutative. There exists a unique
+identity element and inverse element for every element in the set.
+
+Division operation in finite fields is defined as :math:`x / y = x \cdot y^{-1}`,
+that is x multiplied by inverse of y. and substraction :math:`x - y` is defined
+in terms of addition as :math:`x + (-y)` which is x added by negation of y.
+Multiplicative inverse can be *easily* calculated using `extended Euclidean
+algorithm <http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm>`_  which
+I've not understood yet myself as there were readily available library functions
+which does this for us. But I hear from Ramakrishnan that its very easy one.
+
+Elliptic Curve in :math:`\mathbb F_p`
+-------------------------------------
+
+Now we understood what is finite fields we now need to restrict our elliptic
+curves to the finite field. So our original definition of elliptic curve becomes
+slightly different, that is we will have `modulo p` to restrict the elements.
+
+.. math::
+
+   \begin{array}{rcl}
+   \left\{(x, y) \in (\mathbb{F}_p)^2 \right. & \left. | \right. & \left. y^2 \equiv x^3 + ax + b \pmod{p}, \right. \\
+   & & \left. 4a^3 + 27b^2 \not\equiv 0 \pmod{p}\right\}\ \cup\ \left\{0\right\}
+   \end{array}
+
+All our previous operations can now be written as follows
+
+.. math::
+
+   \begin{array}{rcl}
+   x_R & = & (m^2 - x_P - x_Q) \bmod{p} \\
+   y_R & = & [y_P + m(x_R - x_P)] \bmod{p} \\
+   & = & [y_Q + m(x_R - x_Q)] \bmod{p}
+   \end{array}
+
+Where slope, when :math:`P \neq Q`
+
+.. math::
+
+   m = (y_P - y_Q)(x_P - x_Q)^{-1} \bmod{p}
+
+and when :math:`P = Q`
+
+.. math::
+
+   m = (3 x_P^2 + a)(2 y_P)^{-1} \bmod{p}
+
+Couple of difference we see are all operations are now `modulo p` where p is the
+order of finite field.Division operations earlier are all now replaced with
+multiplication with *multiplication inverse*.
+
+Scalar Multiplication and Cyclic Group
+--------------------------------------
+
+Now if we consider scalar multiplication we discussed above with finite fields,
+we get some thing called *cyclic subgroups*. Considering a point `P` in the
+finite field, if we add 2 multiples of point `P` we get another multiple of
+point `P` (i.e. multiples of `P` are closed under addition). Hence set of
+multiples of P is a cyclic subgroup.
+
+.. math::
+
+   nP + mP = \underbrace{P + \cdots + P}_{n\ \text{times}} + \underbrace{P +
+   \cdots + P}_{m\ \text{times}} = (n + m)P