Step 1: Pick ECC curve, parameters and generate key pair

Elliptic curve for bitcoin is always  y^2 = x^3 +ax + b 
Known as spec256k1, established by National Institute of Standards and Technology (NIST)
Domain paramenters (p,a,b,G,n,h) are all set/fixed by Bitcoin
p is prime modulo of field Fp
Finite fields are modular, things wrap around until they fit in
a=0, b=7
G is base point (always the same)
n is order of point G. Smallest positive number n such that nG=infinity
so number of times G can be added to itself to make a vertical line
h = 1, the cofactor = Order of elliptic curve / order of subgroup
By Lagrange, order of a any finite group is divisible by the order of subgroup
so h is always an integer

-Private Key: any number b/w 1 and 2^256 (n=order of curve) - 1
This is randomly generated by feeding a larger string of numbers through SHA256 hash to get a 256-bit number. If less then n-1, then great, if not, then try again.

Order 2^256 is approx. 10^77
No. of atoms in universie is 10^80

-Public key: base point x private key
^Example of a trapdoor function: Discrete log problem
Easy to compute in one direction but nearly impossible to go backwards

Step 2: Hash data using algorithm e.g SHA-1 and sign hashed data
Signature (r,s) is created, different each time
but created from the same private key
Step 3: Verify

If the wallet application does not maintain a copy of unspent transaction outputs, it can query the bitcoin network to retrieve this information, using a variety of APIs.

Blockchain explorer: bitcoin search engine to search for addresses, transactions, blocks