# Half of any bitcoin (crypto) public key - (public key half) is possible

I found a topic on bitcointalk

Public key x and y == Double(Half of the Public key x and y)

half any public key is possible how that possible , in crypto there is subtraction and multiplication only then how divition

one post Sure, just multiply the point by 2^-1 (mod n).

can some one explain more which point to multiply and what is 2^-1 (mod n)

Half of any bitcoin (crypto) public key - (public key half)

Firstly you have an elliptic curve, e.g. Bitcoin uses a Koblitz curve secp256k1 $y^2 = x^3 + 7$.
The group is defined over curve points over a finite field $F_p$ (integer modular $p$). The group elements are points on the curve. A point in the affine form consists of two coordinates $P =(x,y)$ where $x,y\in F_p$.
For group elements, you can do point addition $P+Q$, as well as scalar multiplication $sP$, where $s$ is an integer in $Z_n$ where $n$ is the order of the group (how many elements in the group).
A public key in bitcoin is a point $P$. To do $\frac{P}{2}$, you multiply $\frac{1}{2}$ to $P$ where $\frac{1}{2}$ is the multiplicative inverse of $2$ in $Z_n$. It is an integer that can be found using the extended Euclidean algorithm and is 57896044618658097711785492504343953926418782139537452191302581570759080747169 in the case of secp256k1.
• You are wrong in that 3/2 is not equal to 1 modulo $n$ in this context. It is $3 \cdot 2^{-1} \bmod n$. Be aware we are not using integer arithmetics, but modular arithmetics. – Changyu Dong Jun 12 '18 at 18:11