Posted on Jul 28, 2018 By copyninja under development

TLDR; this post is going to be too long and mostly theoretical. If you are wondering about where is Go in all these posts, please wait for few more posts before I can write about the actual implementation.

In my previous post I talked about starting of my new adventure to write a cryptographic library and how I tried to brute force to solve problem without knowing basics. In this post I'll talk about my learning of Elliptic Curves and their Groups. This post is just my notes and I'm not trying to explain all basics or mathematics behind the Elliptic curves.

Ramakrishnan gave me a good article to start with Elliptic curves. Its a series of posts by Andrea Corbellini, which starts with Elliptic Curve Cryptography: A Gentle Introduction. These posts gave me a great deal of understanding about elliptic curves and how elliptic curve cryptography works, and why its faster. If you want to learn about elliptic curve I highly recommend you to go through the series.

Elliptic Curves

Elliptic curves are set of points described by equation

\begin{equation*} y^2 = x^3 + ax + b \end{equation*}

where a and b are the coefficients which needs to satisfy \(4a^3 + 27b^2 \neq 0\). This form of equation is called Weirstrass normal form of elliptic curves.

Along with normal point there is also point on curve which is considered ideal point and is point at infinity; it is denoted by symbol 0. So considering point at infinity we can define elliptic curve as

\begin{equation*} \{(x,y) \in \mathbb R ^2 | y^2 = x^3 + ax + b, 4a^3 + 27b^2 \neq 0 \} \cup \{0\} \end{equation*}

Groups

Before going to Elliptic curve groups we need to know what are groups. I now remembered my academics where I had studied about groups and number theory in general. That time never knew where exactly it was useful. And I should really say learning something without knowing its application is really difficult. Moving on to the groups,

A group in mathematics is basically a set for which there is a binary operation which is called as "addition" and idicated with + symbol. In order for set \(\mathbb G\) the binary operation must be defined so that it has following properties.

  1. closure: if a and b are members of of \(\mathbb G\) then \(a + b\) is also member of \(\mathbb G\).
  2. associativity: \((a + b) + c = a + (b + c);\)
  3. there exists an identity element 0 such that \(a + 0 = 0 + a = a;\)
  4. Every element has a inverse, that is for every a there exists b such that \(a + b = 0\)

For a group to be Abelian there is a 5th rule.

  1. commutativity: \(a + b = b + a\)

If a group satisfies we get some additional property such that unique identity element and unique inverse element which is either directly or indirectly very important.

Elliptic Curve Groups

Similar to number groups we defined above, its possible to have group over elliptic curves. Following are the property or rule for a Elliptic curve groups.

  1. Elements of the groups are point on elliptic curve;
  2. the identity element is the point at infinity; yes the one we described above while defining elliptic curves.
  3. the inverse of a point \(\mathrm R\) is the one symmetric on x-axis. i.e. if point is \((x,y)\) then inverse of the point is \((x, -y)\)
  4. adition is given by the rules: given three aligned, non-zero points,\(\mathrm P,Q\) and \(\mathrm R\) their sum is\(P + Q + R = 0\). This means the line through the 3 point passes through the point at infinity.
  5. For addition we want 3 points to be aligned and does not require any specific order which means \(P + (Q + R) = Q + (P + R) = R + (P + Q) = ... = 0\) that is our + operator is both associative and commutative

By point 5 we conclude that Elliptic curve groups are Abelian.

Algebraic Addition

Since we know \(P + Q + R = 0\) we can say \(P + Q = -R\). This means we draw a line on elliptic curve which passes through point P and Q it intersects the curve at 3rd point R. Inverse of point R is -R and is the result of addition of 2 points P and Q. This is easy geometrically but for computers, we need to represent this with algebraic notation. This is where the slope of line comes to picture, another academic topic which I learnt without actually knowing its real world application.

So considering point P as \((x_P,y_P)\) and Q as \((x_Q, y_Q)\) we can calculate slope of line going from P to Q as follows

\begin{equation*} m = \frac{y_P - y_Q}{x_P - x_Q} \end{equation*}

And in a special case where \(P = Q\) formula is slightly different

\begin{equation*} m = \frac{3x_P^2 + a}{2y_P} \end{equation*}

Intersection of this line with curve is point R \((x_R,y_R)\) and point \(x_R\) and \(y_R\) are calculated using following formula.

\begin{equation*} x_R = m^2 - x_P - x_Q \end{equation*}
\begin{equation*} y_R = \begin{cases} y_P + m(x_R - x_P), & \text{or} \\ y_Q + m(x_R - x_Q) \end{cases} \end{equation*}

I've not mentioned about special cases of elliptic curve additions which is explained in Andrea Corbellini blog. Please refer for more mathematical oriented explanation.

Scalar Multiplication

One more important operation we do with elliptic curve group is scalar multiplication. It allows you to calculate \(nP\) where n is a natural number. Scalar multiplication can be implemented using addition as follows.

\begin{equation*} nP = \underbrace{P + P + \cdots + P}_{n\ \text{times}} \end{equation*}

This property is what makes elliptic curves attractive and faster than the normal integer groups where scalar multiplication is actually exponentiation operation.

If we see scalar multiplication formula above, we see that it requires n addition, which can be \(O(n)\) operation or, if we consider n as k bit integer then algorithm will be \(O(2^k)\) is large prime number.

But there is another algorithm which is much faster for this purpose, its called double and add you can find wikipedia article on it. First I did not understand this, but after a while of struggling, and trying to implement it myself it became pretty clear. It works with first taking binary representation of n \(n = n_0 + 2n_1 + 2^2_n2 + ...+2^kn_k\) where \([n_0..n_k] \in \{0,1\}\). Below is pseudo code for the recursive version of algorithm.

def scalarmult(P, n):
       if n == 1:
           return P # identity element
       if n & 1:
           # n is odd we add point
           return point_add(P, scalarmult(P, n - 1)
       else:
           return scalarmult(point_double(P), n >> 1) # we double point and multiply by n/2

Let's consider an example to understand what is this algorithm doing. Lets say we want to caculate 21P. First thing is we denote 21 as binary string 10101 now skipping all those 0's and consider only the 1's, so we will have \(2^4 + 2^2 + 2^0\). Now if we multiply P to this representation. So this algorithm calculates 21P as \(16P + 4P + P\).

If we consider doubling and adding as constant operations i.e. \(O(1)\) then this algorithm will be \(O(\log n)\) (in above 21 can be calculated in 3 operations which is \(\log 21\)) which is much better than earlier \(O(n)\) algorithm.

Given n and P we can easily cacluate \(Q = nP\) but given Q and P there is no easy way to find n especially when numbers are big. This is the "hard" problem or the discrete logarithm problem which makes cryptography work.

Conclusion

So I learned now about Elliptic curves, its group and operations like scalar multiplication and addition. But this is not directly useful, we need to learn about Finite Fields and Elliptic Curve over Finite Field and many more concepts before we can go to actual implementation!. I thought of writing finite field as part of this post, but then I've already written a lot and there is no point in expanding this post more. So finite fields, cyclic groups will be part of next post.

As a closing note, if you are wondering why there is no mention of anything about Go, you need to wait a bit more for my actual implementation notes in Golang. I would like to write down entire learning process rather than directly jumping to implementation. So please bare with me :-).