The Lablans Conjecture: Enumeration of Structural Field Implementations over Finite Sets

Abstract

This document introduces the Lablans Conjecture, which posits that for a finite field 𝔽n = GF(n), there exist exactly n! distinct structural implementations of the field over a fixed set of n elements. These implementations differ in their concrete operation tables (addition and multiplication) due to permutations of element labels and associated shifts in identity elements and operation structure. Each such implementation satisfies the field axioms and is isomorphic to GF(n), but they are not structurally identical.

1. Introduction

In classical field theory, it is standard to treat all finite fields of the same order as isomorphic, abstracting away from specific element labels or operation tables. However, practical implementations in cryptography, hardware design, and machine learning often work over fixed element sets (e.g., {0,1,…,n-1}) and rely on explicit binary operation tables.

This paper investigates the number of structurally distinct implementations of GF(n) over such fixed sets, where "structurally distinct" means differing in the assignment of the additive and multiplicative identity elements, as well as the corresponding full operation tables. These are not merely isomorphic via renaming; they represent different algebraic mappings over the same base set.

2. The Lablans Conjecture

Conjecture (Lablans): Let S = {s0, s1, …, sn-1} be a fixed set of n elements. Then, the number of structurally distinct binary operation table pairs (+, ⋅) on S × S that satisfy the field axioms and define a field isomorphic to GF(n) is exactly n!.

These n! versions arise from all possible permutations of the field elements, which result in different locations of identity elements and hence different looking tables. Each implementation uses the same underlying group structure (additive and multiplicative) but represented under a different bijective map.

Note: All such fields are distinct as finite fields, even if they share a single function; each has a different corresponding function—what we prefer to call the "law of composition" of the finite field.

3. Definitions

• A structural implementation of GF(n) over a set S is a pair of binary operations +, ⋅ : S × S → S such that:
  1. (S, +) is an abelian group with identity e+.
  2. (S \ {e+}, ⋅) is an abelian group with identity e⋅.
  3. ⋅ distributes over +.
• Two implementations are considered structurally distinct if their operation tables differ under the fixed labeling of elements in S.

4. Empirical Verification for n = 2 and n = 3

• For n = 2: There are 2! = 2 such implementations.
• For n = 3: One can construct 6 different field tables over {0,1,2}, each corresponding to a permutation of the element labels.

Each permutation leads to a different placement of the additive and multiplicative identities, changing the apparent form of the operation tables even though all implementations remain algebraically isomorphic to GF(3).

5. Implications and Applications

• In cryptographic primitives, table-based field implementations may leak structure via side-channels.
• Hardware designs may exploit or enforce specific structural forms of fields.
• The Lablans Conjecture allows one to explore the full design space of concrete field implementations without exiting the abstract field class.

6. Future Work

• General proof for arbitrary prime n.
• Enumeration and classification of structural implementations for GF(pk) where k > 1.
• Study of automorphism-fixed implementations and equivalence classes under inner automorphisms.

Acknowledgment

This document formalizes the intuition and empirical observation by the proposer that implementation diversity over fixed sets deserves mathematical treatment. We name this result the Lablans Conjecture in recognition of its original formulation.