Preface: Why the Sum-Space Conjecture?

The Sum-Space Conjecture follows a conceptual lineage similar to the Lablans Conjecture, which challenged the assumption that a finite field representation over a given prime \( n \) is unique. That earlier conjecture, supported by the construction of the Finite Lab Transform (FLT), revealed that \( n! \) distinct field representations exist — each arising from a permutation of the base element set.

Background and Motivation

In classical computing and cryptography, addition is performed in fixed-radix systems (binary, decimal, etc.) using deterministic ripple-carry logic. For example, ChaCha20 uses modulo-\(2^{32}\) addition for key mixing and round operations.

This work generalizes digit-wise addition to allow arbitrary carry functions while maintaining a reversible residue function. The goal is to preserve statistical uniformity — a property critical to cryptographic strength, where bias in ciphertext can lead to vulnerability.

Generalized Addition Model

Let \(n \geq 2\) be a fixed integer radix. We define the augmented word vector at time \(t\) as: \[a^{(t)} = [a_1^{(t)}, a_2^{(t)}, \dots, a_k^{(t)}], \quad \text{(augend)}\] and the corresponding addend: \[b^{(t)} = [b_1^{(t)}, b_2^{(t)}, \dots, b_k^{(t)}], \quad \text{(addend at } t=0\text{)}\] Each element \(a_i^{(t)}, b_i^{(t)} \in \{0, \dots, n-1\}\).

Define two functions:

• A residue function \(\text{res}: \{0, \dots, n-1\} \times \{0, \dots, k_i\} \to \{0, \dots, n-1\}\)

• A carry function \(\text{car}: \{0, \dots, n-1\} \times \{0, \dots, k_i\} \to \{0, \dots, k_i\}\)

Then the recurrence relations for each step \(t\) and position \(p\) are: \[r_p^{(t)} = \text{res}(r_p^{(t-1)}, c_p^{(t-1)})\] \[c_p^{(t)} = \begin{cases} \text{car}(r_{p-1}^{(t-1)}, c_{p-1}^{(t-1)}), & \text{if } t+p \leq k \\ z, & \text{otherwise} \end{cases}\] The iteration proceeds until \(t = k\). The initial conditions are: \[r_p^{(0)} = a_p^{(0)} + b_p^{(0)} \pmod{n}, \quad c_0^{(0)} = z\]

This models a dynamic, recursive carry propagation structure with data-dependent behavior at each position and step.

The Sum-Space Conjecture

Computational Evidence

Experiments were conducted for \(n = 2\) to \(256\) and \(m = 2\) to \(8\). Carry functions included random, asymmetric, and position-based definitions. In all cases with bijective residue functions, the output distribution was flat.

One test used the following GF(4) addition matrix as \(r\): \[r(a, b) = \begin{bmatrix} 0 & 1 & 2 & 3 \\ 1 & 0 & 3 & 2 \\ 2 & 3 & 0 & 1 \\ 3 & 2 & 1 & 0 \\ \end{bmatrix}\]

With a random \(4 \times 4\) carry table: \[c(a,b) = \begin{bmatrix} 1 & 3 & 0 & 3 \\ 0 & 3 & 3 & 0 \\ 1 & 0 & 3 & 2 \\ 0 & 2 & 0 & 3 \\ \end{bmatrix}\]

Each of the \(256\) output vectors in \(\{0,1,2,3\}^4\) appeared exactly once across \(65,536\) input pairs. No statistical bias was found.

Cryptographic Implications

• Key mixing in AES-like systems may adopt structured carry logic instead of XOR.
• Keystream generation may utilize randomized carry propagation without reversibility.

Discussion and Open Problems

Reversibility via Flipped Carry Tables

Let the same residue function be used for addition and subtraction. A forward carry table and a flipped (borrow) table may define inverse operations.

For example, with the following carry and borrow tables:

\[\text{Carry} = \begin{bmatrix} 0 & 0 & 0 & 0 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 2 & 2 \\ 0 & 1 & 2 & 3 \\ \end{bmatrix} \quad \text{Borrow} = \begin{bmatrix} 0 & 1 & 2 & 3 \\ 0 & 0 & 2 & 2 \\ 0 & 1 & 0 & 1 \\ 0 & 0 & 0 & 0 \\ \end{bmatrix}\]

This allows fully invertible word-wise operations, even in cryptographic modes that typically use XOR (which lacks carry propagation).

Recursive Update Dynamics in Sum Spaces

Define: \[s^{(t+1)}_i = r(s^{(t)}_i + \gamma^{(t)}_i), \quad \gamma^{(t+1)}_i = c_i(s^{(t)}_{i-1}, \gamma^{(t)}_{i-1})\] with boundary condition \(\gamma^{(t)}_0 = z\).

This enables time-recursive, reversible, radix-generalized arithmetic sequences.

Transform With FLT

One now can apply the earlier mentioned FLT [2] to transform both the residue and carry/borrow generating n-state functions.

This further adds to the immense solution space of modified computational implementation.

Usage and Patent Notice

The Finite Lab-Transform (FLT) and related concepts are protected under issued USA patents. Use for educational, review, research, and testing is permitted. Operational or commercial use requires proper licensing.

References

1. Peter Lablans. The Lablans Conjecture: Enumeration of Structural Field Implementations over Finite Sets, 2025. https://lablansconjecture.com
2. Peter Lablans. Website: Finite Lab-Transform (FLT). https://labtransform.com, 2022.