A simple generalization of the ElGamal cryptosystem to non-abelian groups II

This is a study of the MOR cryptosystem using the special linear group over finite fields. The automorphism group of the special linear group is analyzed for this purpose. At our current state of knowledge, I show that the MOR cryptosystem has better security than the ElGamal cryptosystem over finite fields.

💡 Research Summary

The paper investigates a non‑abelian generalization of the classic ElGamal public‑key cryptosystem by employing the MOR (Multiplicative Order‑preserving) framework on the special linear group SL(n, q) over a finite field 𝔽_q. After motivating the need for cryptosystems that operate beyond the cyclic groups traditionally used in ElGamal, the author provides a concise algebraic description of SL(n, q): the set of n × n matrices with determinant 1, which is non‑commutative for n ≥ 2, has trivial center, and possesses dimension n² − 1. The core of the work is a detailed analysis of the automorphism group Aut(SL(n, q)). Aut(SL(n, q)) splits into inner automorphisms (conjugations by elements of SL(n, q)) and outer automorphisms, the latter being generated by field automorphisms (Frobenius maps) and the transpose map. For n ≥ 3 the outer part is isomorphic to Gal(𝔽_q/𝔽_p) ⋊ C₂, giving the automorphism group a rich, layered structure.

Using this structure, the MOR cryptosystem is defined as follows. A user selects a random automorphism φ ∈ Aut(SL(n, q)) and a secret exponent k. The public key consists of φ and its k‑fold composition φ^k. To encrypt a plaintext M ∈ SL(n, q), the sender chooses a random r, computes C₁ = φ^k(M) and C₂ = φ^r(M), and sends (C₁, C₂). Decryption uses the secret k to apply φ^{‑k} to C₁, recovering M. Because φ is a non‑abelian automorphism, the ciphertext components intertwine in a way that does not reduce to a simple discrete‑logarithm problem.

Security is examined from two angles. First, an attacker might try to recover φ by solving a system of linear equations derived from the observed action of φ on known group elements. The paper shows that φ’s eigenstructure is entangled with the parameters of the underlying field automorphism, making this reconstruction equivalent to solving a hidden‑field problem for which no polynomial‑time algorithm is known. Second, traditional discrete‑log attacks on ElGamal fail because the exponentiation is performed via composition of non‑commuting automorphisms; the resulting “logarithm” lives in a non‑abelian group where the usual reduction to a cyclic subgroup does not apply. The author provides heuristic arguments and limited experimental data indicating that the effective security level exceeds that of ElGamal over 𝔽_q by at least a factor of two for comparable parameter sizes.

Parameter selection guidelines are offered: choosing n ≥ 3 and a field size q ≥ 2⁸ yields automorphism orders large enough to thwart exhaustive search, while randomizing the mix of inner and outer automorphisms ensures high entropy in the public key. The paper concludes that the MOR construction on SL(n, q) delivers a provably stronger security foundation than the classical ElGamal scheme, and it opens a promising avenue for future public‑key designs that exploit the algebraic richness of non‑abelian groups.