We already saw the basic concepts behind the creation of keys and address creation. Let’s try to implement them. I’m going to use Python because it’s easier to follow.

Following the Standards for Efficient Cryptography, we define the constants. In the document, please look for “Recommended 256-bit Elliptic Curve Domain Parameters over Fp“ (secp256k1)

``````p = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
order = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
# Remember the equation we are using is `y^2 % p == x^3 + 7
a = 0x0000000000000000000000000000000000000000000000000000000000000000
b = 0x0000000000000000000000000000000000000000000000000000000000000007
# This is the generator point defined in the standard
g = (0x79BE667EF9DCBBAC55A06295CE870B07029BFCDB2DCE28D959F2815B16F81798,
``````

For pedagogical reasons, I’ve chosen python-ecdsa. It’s a pure python library, really easy to understand. In production settings, it might be worth to explore C alternatives for speed, but more importantly, this library should NOT be used in systems where `os.urandom()` is weak. Once installed the library, and imported to the Python unit, we can create a curve like this

``````# First, let's define the curve
elliptic_curve_secp256k1 = ecdsa.ellipticcurve.CurveFp(p, a, b)
# And, from the curve, extract the generator
generator_secp256k1 = ecdsa.ellipticcurve.Point(elliptic_curve_secp256k1, g, g, order)
# Now, we need to define the object identifiers for the elliptic curve domain parameters
certicom_arc_oid = (1, 3, 132, 0, 10)
#Finally, the curve
secp256k1 = ecdsa.curves.Curve("SECP256k1", elliptic_curve_secp256k1, generator_secp256k1, certicom_arc_oid)
``````

As we know, the private key is only a random number in the range [0, 2^256], we can pick it with a simple python function. The key here is to grab as many entropy as possible for the random draft. the `os.urandom` function is described in the documentation as capable of “Return a string of n random bytes suitable for cryptographic use”. The documentation also points to SystemRandom as an easy-to-use interface.

``````def private_key():
key = sum([(SystemRandom().randrange(2) * 2 ** idx) for idx in range(256)])
return key
``````

Once we have the secret key, we compute the public key by performing the multiplication. Remember, the public key is a point in the curve, in other words, it’s an ordered pair.

``````public_key_point = secret_key * generator_secp256k1
``````

When exposed, the public key is prefixed with `04`, and then, the value of the x and y coordinates expressed as hex numbers:

``````04[hex(x)hex(y)]
``````

But, as discussed before, since you can retrieve the value of `y` given `x` (and the sign of `y`), the compressed format is gaining popularity all the time. The sign of y is indicated by the prefixes `02` (positive ~ even) or 03 (negative ~ odd).

``````def produce_public_compressed_key_from_point(point):
point_y_is_odd = point.y() & 1
if point_y_is_odd:
key = '03' + '%064x' % point.x()
else:
key = '02' + '%064x' % point.x()
return key
``````

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