get x and y components of ec public key using openssl

Question

I am generating a KeyPair for ECC from curve 'secp128r1' using openssl

Steps I followed :

• first I generated a private key using the command

  openssl ecparam -genkey -name secp128r1 -noout -out private.pem

• then i viewed the corresponding public key using the command

  openssl ec -in private.pem -text -noout

  which showed an output as :

  read EC key

  Private-Key: (128 bit)
  priv:
  00:9f:bf:2b:bd:06:86:3a:a1:bc:7c:3e:90:57:40:
  f4:bc
  pub:
  04:04:ce:24:34:d4:cb:f2:58:94:2f:8a:5f:06:d7:
  3f:ed:5a:50:ef:fb:cc:b7:49:62:16:62:9e:aa:d5:
  30:a8:a5
  
  ASN1 OID: secp128r1

I want explicitly x and y components from the public key generated here, please can anyone suggest the correct way of doing this ?
The above public key is 264 bits long, hence cannot take(/split) it as is
Thanks

Answer 1

First of all, secp128r1 is outdated. Use curves that give better security for today's standard. See safecurves by Daniel J. Bernstein and Tanja Lange.

An Elliptic Curve defined over a field of size q and every element -point- has two coordinates X and Y. The Elliptic Curve Secp128r1 has 2¹²⁸-2⁹⁷-1 size ℓ, i.e. number of points a little under 2^128. This means that we need 128-bit representation.

The public key which is also a point on the curve has two coordinates, therefore, we need to store two 128-bit.

If we look at the equation of the elliptic curve Y² = X³ + aX + b where

 a = FFFFFFFD FFFFFFFF FFFFFFFF FFFFFF
 b = E87579C1 1079F43D D824993C 2CEE5E

if we know X from the equation we can find Y. Since we are working in a field the Y can have at most two square roots. Y² will have y or -y as the square root. This knowledge can be used to compress the representation of a point and it is called point compression. Just x coordinate and one bit to select y or -y. Now look at the base point (see Certicom recommendation)

 base point = 03 161FF752 8B899B2D 0C28607C A52C5B86
            = 04 161FF752 8B899B2D 0C28607C A52C5B86 CF5AC839 5BAFEB13 C02DA292 DDED7A83 

The first octet determines the structure

• 04 means there is no compression
• 03 means there is a compression and select y as positive
• 02 means there is a compression and select y as negative

---

Now turn into OP's parameters;

pub:
04:
04:ce:24:34:d4:cb:f2:58:94:2f:8a:5f:06:d7:3f:ed: -->X
5a:50:ef:fb:cc:b7:49:62:16:62:9e:aa:d5:30:a8:a5  -->Y

first octet 04 means there is no compression. The first line is the X coordinate and the second line is the Y coordinate of your public key.

What about the private key n? It is just a scalar -integer- between 0<=n<=ℓ

priv: 00:9f:bf:2b:bd:06:86:3a:a1:bc:7c:3e:90:57:40:f4:bc

Therefore, the above number - not point - is your private key.

You can also use some web tools to extract this information.

• 8gwifi.org - Crypto Playground

Note: please don't expose your private key.