Ternary Diffie-Hellman (TDH) Key Exchange

Ternary Diffie-Hellman (TDH) is an adaptation of the classic Diffie-Hellman key exchange protocol to work with balanced ternary logic. It operates in a ternary finite field (GF(3^n)) and uses modular arithmetic for secure key exchange.

Algorithm Outline

Step 1: Shared Parameters

Choose a large balanced ternary prime p and a primitive ternary generator g.

\[g \text{ generates all non-zero elements of } GF(3^n) \text{ modulo } p\]

Balanced ternary primes are derived from equations of the form 3^n + k, where k satisfies primality conditions.

Step 2: Private Key Generation

Party A chooses private key a
Party B chooses private key b

Step 3: Public Key Generation

Both parties compute their public keys using balanced ternary modular exponentiation:

\[A_{pub} = g^a \mod p\]\[B_{pub} = g^b \mod p\]

Step 4: Shared Secret Computation

Each party computes the shared secret:

\[S_A = (B_{pub})^a \mod p\]\[S_B = (A_{pub})^b \mod p\]

Due to the properties of modular arithmetic, S_A = S_B = g^(ab) mod p, which becomes the shared secret.

Advantages of Balanced Ternary Diffie-Hellman

Increased Keyspace: Operating in GF(3^n) provides a larger keyspace compared to binary fields of the same size.
More Secure Generators: Balanced ternary's complexity in modular exponentiation increases resistance to attacks like the Index Calculus Method.
Better Efficiency in Native Ternary Systems: Native ternary hardware could process g^a mod p faster due to the balanced ternary representation.

Challenges

Lack of optimized ternary hardware: Current binary machines emulate ternary, potentially reducing performance.
Primality testing in ternary systems is more computationally expensive than binary.

Example Implementation

Here's a basic implementation of Ternary Diffie-Hellman in Python:

import random

def ternary_mod_pow(base, exponent, modulus):
    result = 1
    base = base % modulus
    while exponent > 0:
        if exponent % 3 == 1:
            result = (result * base) % modulus
        elif exponent % 3 == 2:
            result = (result * base * base) % modulus
        exponent = exponent // 3
        base = (base * base * base) % modulus
    return result

def generate_ternary_prime(bits):
    # This is a placeholder. In practice, use a robust primality test.
    return random.getrandbits(bits) | 1

# TDH parameters
p = generate_ternary_prime(1024)  # Large ternary prime
g = 3  # Generator

# Alice's keys
a_private = random.getrandbits(256)
a_public = ternary_mod_pow(g, a_private, p)

# Bob's keys
b_private = random.getrandbits(256)
b_public = ternary_mod_pow(g, b_private, p)

# Shared secret computation
alice_secret = ternary_mod_pow(b_public, a_private, p)
bob_secret = ternary_mod_pow(a_public, b_private, p)

assert alice_secret == bob_secret, "Shared secrets do not match!"
print("Shared secret established successfully!")

Ternary Key Exchange