Ternary Elliptic Curve Cryptography (TECC)

Ternary Elliptic Curve Cryptography (TECC) is an adaptation of traditional Elliptic Curve Cryptography to work with balanced ternary logic. It 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
  • a, b: Constants in the ternary field satisfying 4a^3 + 27b^2 ≠ 0 (to ensure the curve has no singularities)
  • Points (x, y) lie on the curve in GF(3^n)

Algorithm Outline

Step 1: Shared Curve Parameters

  1. Both parties agree on a ternary elliptic curve E and a base point G (a point on the curve of large order).
  2. The curve and G are public.

Step 2: Key Generation

  • Party A chooses a private key a (a random ternary number).
  • Party B chooses a private key b (another random ternary number).
  • Both compute their public keys:\[A_{pub} = aG\]\[B_{pub} = bG\](scalar multiplication of G by a and b respectively)

Step 3: Shared Secret

  • Party A computes S_A = aB_pub = abG
  • Party B computes S_B = bA_pub = abG
  • The shared secret is S = abG, a point on the curve.

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. For 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.