Conceptual adaptor signature (not production code):

# Conceptual adaptor signature (not production code):

def create_adaptor_signature(secret_key, message, adaptor_point):
    """
    adaptor_point: T = t × G, the public commitment to the secret t
    """
    # Standard Schnorr nonce
    k = generate_nonce(secret_key, message)
    R = k * G  # Nonce point

    # Shift the nonce point by the adaptor point
    R_prime = R + adaptor_point  # R' = R + T

    # Compute adaptor signature (like Schnorr but with shifted R)
    e = tagged_hash("BIP0340/challenge", 
                    x_only(R_prime) + x_only(secret_key * G) + message)

    s_prime = k + int.from_bytes(e, 'big') * secret_key  # s' = k + e*x
    # Note: NOT s' = k + t + e*x (the secret t is NOT included)

    return (R_prime, s_prime)  # Adaptor signature

def complete_adaptor_signature(adaptor_sig, secret_t):
    """Complete the adaptor signature by adding the secret."""
    R_prime, s_prime = adaptor_sig
    s = s_prime + secret_t  # s = s' + t
    # Now (R_prime, s) is a valid Schnorr signature
    return (R_prime, s)

def extract_secret(adaptor_sig, completed_sig):
    """Extract the secret from an adaptor and its completion."""
    R_prime, s_prime = adaptor_sig
    R_prime, s = completed_sig
    t = s - s_prime  # t = s - s'
    return t

Traditional HTLC atomic swap (on-chain trace):
Alice's chain: OP_SHA256 <hash_of_secret> OP_EQUAL ← secret visible when Bob claims
Bob's chain:   OP_SHA256 <hash_of_secret> OP_EQUAL ← links the two transactions

Adaptor signature atomic swap:
Alice's chain: Key path spend (single Schnorr sig) ← looks like normal transaction
Bob's chain:   Key path spend (single Schnorr sig) ← looks like normal transaction
No on-chain link between the two swaps

Current Lightning (HTLC):
Payment path: Alice → Bob → Carol → Dave
Each hop reveals the same hash preimage
→ Hops can correlate payments, privacy is poor

Future Lightning (PTLC with adaptor signatures):
Payment path: Alice → Bob → Carol → Dave
Each hop uses a DIFFERENT adaptor point (related by a tweak)
→ Hops cannot correlate payments (each sees a different secret)
→ Payment privacy is massively improved