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What’s The Order in ECC?

I love using elliptic curve cryptography (ECC), but there’s one little parameter of the curve that’s not so easy to explain … the order (n). Let me first give you a bluffers guide to ECC …

• We have a curve y²=x³+ax+b (mod p), and where a and b define the curve and p is a prime number. An example is where a=0 and b=7, and which gives y²=x³+7 (mod p).
• We select a base point of the curve (G) and then make all the operations in relation to this.
• First, we generate a random number for our private key (a), and then we calculate xG (mod p), and which is the point G, added x times (with the mod p operation). The value of a is the private key, and xG (mod p) is the public key.

All fine up to now? And so we have p, a, b, and G:

EllipticCurve = collections.namedtuple('EllipticCurve', 'name p a b g n h')

curve = EllipticCurve(
    'secp256k1',
    # Field characteristic.
    p=0xfffffffffffffffffffffffffffffffffffffffffffffffffffffffefffffc2f,
    # Curve coefficients.
    a=0,
    b=7,
    # Base point.
    g=(0x79be667ef9dcbbac55a06295ce870b07029bfcdb2dce28d959f2815b16f81798,
       0x483ada7726a3c4655da4fbfc0e1108a8fd17b448a68554199c47d08ffb10d4b8),
    # Subgroup order.
    n=0xfffffffffffffffffffffffffffffffebaaedce6af48a03bbfd25e8cd0364141,
    # Subgroup cofactor.
    h=1,
)