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diff --git a/content/development/golang_spake2_ecc2.rst b/content/development/golang_spake2_ecc2.rst index 16ed395..4897338 100644 --- a/content/development/golang_spake2_ecc2.rst +++ b/content/development/golang_spake2_ecc2.rst @@ -1,11 +1,10 @@ SPAKE2 In Golang: Finite fields of Elliptic Curve ################################################# -:date: 2018-07-29 19:02 +0530 +:date: 2018-08-12 22:51 +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. @@ -135,8 +134,61 @@ Subgroup Order Subgroup order tells how many points are really there in the subgroup. We can redefine the *order of group* in subgroup context as **order of P is the -smallest positive integer such that nP = 0**. Order of subgroup is linked to -order of elliptic curve by `Lagrange's Theorem +smallest positive integer such that nP = 0**. In above case if you see we have +smallest `n` as `5` since `5P = 0`. So order of subgroup above is 5, it contains +5 element. + +Order of subgroup is linked to order of elliptic curve by `Lagrange's Theorem <https://en.wikipedia.org/wiki/Lagrange%27s_theorem_(group_theory)>`_ which says **the order of subgroup is divisor of order of parent group**. Lagrange is another name which I had read in my college, but the algorithms were different. + +From this we have following steps to find out the order of subgroup with base +point `P` + +1. Calculate the elliptic curve's order `N` using Schoof's algorithm. +2. Find out all divisors of `N`. +3. For every divisor of `n`, compute `nP`. +4. The smallest `n` such that `nP = 0` is the order of subgroup `N`. + +Note that its important to choose smallest divisor, not a random one. In above +examples 5P, 10P, 15P all satisfy condition but order of subgroup is 5. + +Finding Base Point +------------------ + +Far all above which is used in ECC, i.e. Group, subgroup and order we need a +base point `P` to work with. So base point calculation is not done at the +beginning but in the end i.e. first choose a order which looks good then look +for subgroup order and finally find the suitable base point. + +We learnt above that subgroup order is divisor of group order which is derived +from *Lagrange's Theorem*. This term :math:`h = N/n` is actually called +**co-factor of the subgroup**. Now why is this term co-factor important?. +Without going into details, this co-factor is used to find generator for the +subgroup as :math:`G = hP`. + +Conclusion +=========== + +So now are you wondering why I went on such length to describe all these?. Well +one thing I wanted to make some notes for myself because you can't find all +these information in single place, another these topics we talked in my previous +post and this point forms the domain parameters of *Elliptic Curve +Cryptography*. + +Domain parameters in ECC are the parameters which are known publicly to every +one. Following are 6 parameters + +* Prime `p` which is order of Finite field +* Co-efficients of curve `a` and `b` +* Base point :math:`\mathbb G` the generator which is the base point of curve + that generates subgroup +* Order of subgroup `n` +* Co-factor `h` + +So in short following is the domain parameters of ECC :math:`(p, a, b, G, n, h)` + +In my next post I will try to talk about the specific curve group which is used +in SPAKE2 implementation called **twisted Edwards curve** and give a brief +overview of SPAKE2 protocol. |