require 'openssl'

group = OpenSSL::PKey::EC::Group.new('prime256v1')
generator = group.generator #the generator point

ec = OpenSSL::PKey::EC.new
ec.group = group
ec.generate_key #generate a key pair

priv = ec.private_key #this is a random number, a OpenSSL::BN

# the public point P is given by P = priv * G,
# where s is the private key and G is the generator point
pub_point = generator.mul(priv) 

puts ec.public_key == pub_point # true, now convert the point to an EC

pub_ec = OpenSSL::PKey::EC.new
pub_ec.group = group
pub_ec.public_key = pub_point

puts ec.to_text
puts pub_ec.to_text #pub parts are equal!

puts pub_ec.to_pem #to serialize it in PEM format