How can you extract secret keys from ECDSA by doing: d = (((s1 * k) - h1) * r^-1) % n and how RFC6979 fix it

secp256k1_key_recover.sage

# Import needed libraries
import hashlib

# secp256k1 parameters
p  = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
a  = 0
b  = 7
n  = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
Gx = 55066263022277343669578718895168534326250603453777594175500187360389116729240
Gy = 32670510020758816978083085130507043184471273380659243275938904335757337482424

# Define curve over F_p
F = FiniteField(p)
E = EllipticCurve(F, [a, b])
G = E(F(Gx), F(Gy))

# Generate a private key
d = randint(1, n-1)
Q = d * G  # Public key

# Messages
m1 = b"Message One"
m2 = b"Message Two"

# Hash messages (SHA-256)
H1 = Integer(hashlib.sha256(m1).hexdigest(), 16) % n
H2 = Integer(hashlib.sha256(m2).hexdigest(), 16) % n

# Use same nonce (bad practice)
k = randint(1, n-1)
R = k * G
r = Integer(R.xy()[0]) % n

# Signature generation
s1 = (inverse_mod(k, n) * (H1 + d * r)) % n
s2 = (inverse_mod(k, n) * (H2 + d * r)) % n

# Attacker's recovery of private key from reused nonce
numerator = (s1 * H2 - s2 * H1) % n
denominator = (r * (s2 - s1)) % n
d_recovered = (numerator * inverse_mod(denominator, n)) % n

# Output
print("Original private key: ", d)
print("Recovered private key:", d_recovered)
print("Match:", d == d_recovered)

secp256k1_rfc6979.sage

import hashlib
import hmac

# secp256k1 parameters
p  = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEFFFFFC2F
a  = 0
b  = 7
n  = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
Gx = 55066263022277343669578718895168534326250603453777594175500187360389116729240
Gy = 32670510020758816978083085130507043184471273380659243275938904335757337482424

# Define curve
F = FiniteField(p)
E = EllipticCurve(F, [a, b])
G = E(Gx, Gy)

# Helper: RFC 6979 deterministic k generation
def rfc6979_generate_k(privkey, hashmsg, q):
    hlen = 32
    bx = privkey.to_bytes(32, 'big') + hashmsg.to_bytes(32, 'big')
    v = b'\x01' * hlen
    k = b'\x00' * hlen

    k = hmac.new(k, v + b'\x00' + bx, hashlib.sha256).digest()
    v = hmac.new(k, v, hashlib.sha256).digest()
    k = hmac.new(k, v + b'\x01' + bx, hashlib.sha256).digest()
    v = hmac.new(k, v, hashlib.sha256).digest()

    while True:
        v = hmac.new(k, v, hashlib.sha256).digest()
        k_candidate = Integer(int.from_bytes(v, 'big'))
        if 1 <= k_candidate < q:
            return k_candidate
        k = hmac.new(k, v + b'\x00', hashlib.sha256).digest()
        v = hmac.new(k, v, hashlib.sha256).digest()

# Sign a message deterministically
def ecdsa_sign_rfc6979(msg_bytes, privkey):
    z = Integer(hashlib.sha256(msg_bytes).hexdigest(), 16) % n
    k = rfc6979_generate_k(privkey, z, n)
    R = k * G
    r = Integer(R.xy()[0]) % n
    s = (inverse_mod(k, n) * (z + r * privkey)) % n
    return (r, s, k)

# Generate private key
d = randint(1, n - 1)
print("Private key d:", d)

# Sign two messages with RFC6979 (deterministic k)
m1 = b"Message One"
m2 = b"Message Two"

r1, s1, k1 = ecdsa_sign_rfc6979(m1, d)
r2, s2, k2 = ecdsa_sign_rfc6979(m2, d)

print("\nSignature 1:")
print(" r1 =", r1)
print(" s1 =", s1)
print(" k1 =", k1)

print("\nSignature 2:")
print(" r2 =", r2)
print(" s2 =", s2)
print(" k2 =", k2)

print("\nReused nonce?", k1 == k2)