Designing a Ternary Key Exchange

To adapt Diffie-Hellman (DH) and Elliptic Curve Cryptography (ECC) to balanced ternary logic, the fundamental goal remains the same: securely establish a shared secret between two parties over an insecure channel. However, using balanced ternary logic introduces unique properties and challenges.

Here's how a Ternary Diffie-Hellman (TDH) and Ternary Elliptic Curve Cryptography (TECC) might be designed.

1. Ternary Diffie-Hellman (TDH) Key Exchange

Diffie-Hellman relies on modular arithmetic and exponentiation over a finite field. In a balanced ternary adaptation, the cryptosystem operates in a ternary finite field (\[GF(3^n)\]).

Algorithm Outline

Advantages of Balanced Ternary Diffie-Hellman

  1. Increased Keyspace: Operating in \[GF(3^n)\] provides a larger keyspace compared to binary fields of the same size.
  2. More Secure Generators: Balanced ternary's complexity in modular exponentiation increases resistance to attacks like the Index Calculus Method (used in binary DH).
  3. 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.

2. Ternary Elliptic Curve Cryptography (TECC)

Elliptic Curve Cryptography relies on the mathematical properties of elliptic curves defined over finite fields. A ternary adaptation involves defining elliptic curves over a ternary finite field (\[GF(3^n)\]).

Elliptic Curve Definition in Ternary

An elliptic curve in balanced ternary can be defined as:

\[y^2 = x^3 + ax + b \mod p\]

Where:

  • \[p\]: A large ternary prime (as in TDH).
  • \[a, b\]: Constants in the ternary field satisfying \[4a^3 + 27b^2 \neq 0\] (to ensure the curve has no singularities).
  • Points \[(x, y)\] lie on the curve in \[GF(3^n)\].

Algorithm Outline

Advantages of Ternary ECC

  1. Smaller Key Sizes: TECC offers the same level of security as binary ECC with smaller key sizes due to the larger ternary keyspace.
    • Example: A 192-bit binary ECC key might correspond to a 128-trit TECC key.
  2. High Computational Efficiency: Ternary elliptic curve arithmetic (addition and multiplication) can potentially be faster on ternary-native systems.
  3. Quantum Resistance: Larger entropy in ternary systems may increase resistance to quantum attacks, though this needs further research.

Challenges

  1. Curve Security Testing: Balanced ternary elliptic curves need rigorous testing to ensure resistance against attacks like MOV reduction or small-subgroup attacks.
  2. Implementation Overhead: Developing scalar multiplication algorithms for ternary elliptic curves requires new methods adapted for balanced ternary arithmetic.
  3. Hardware Bottleneck: Efficient implementation depends on the availability of ternary processors.

Future Steps

  1. Develop Efficient Modular Arithmetic Algorithms:
    • Focus on ternary modular addition, multiplication, and inversion, as these form the foundation of TDH and TECC.
  2. Create Ternary Field Arithmetic Libraries:
    • Develop libraries for elliptic curve arithmetic in \[GF(3^n)\].
  3. Simulate on Binary Hardware:
    • Emulate ternary systems on binary hardware for initial testing.
  4. Test for Quantum Resistance:
    • Investigate the resilience of TDH and TECC against quantum algorithms.

Summary

Ternary Diffie-Hellman and Elliptic Curve Cryptography offer significant theoretical advantages in keyspace size, computational efficiency, and security. While challenges like hardware optimization and algorithmic testing remain, they represent a promising direction for future cryptographic innovation.