RESEARCH REPORT / 07 SEPTEMBER 2026

Decompwlj & cryptography

Exact connections, plausible research directions, and the evidence needed to move from structure to an algorithm.

Assessment. Decomposition into weight × level + jump (WLJ) connects cryptography-relevant objects: divisors, neighboring integers, and residue classes. Its clearest immediate value is as an arithmetic research lens. An exact reduction below links general WLJ computation to integer factorization. This establishes a computational connection, but does not supply a faster factoring algorithm or demonstrate a weakness in deployed cryptography.

1 / The mathematical object

For a strictly increasing integer sequence, write a = a(n), b = a(n+1), and d = b − a. If a > 2d, define

l = a − d,   k = min{t ∈ ℕ : t > d and t ∣ l},   L = l/k.

a = kL + d,   with   0 < d < k.

Otherwise l = k = L = 0 and the term is unclassified; the displayed decomposition identity then need not hold. The existence condition is b < 3a/2. A decomposable term is level-classified if k > L, and weight-classified if k ≤ L, including ties. The successor matters: WLJ is attached to a sequence, not to an isolated integer. [1, 2]

A small, independently checked sample

These examples exhibit both classes and composite weights. In particular, k is not always prime. All entries were checked directly by divisor enumeration; they are illustrations, not a cryptographic benchmark.

Source and proof status

The available treatise is the eighth edition, dated 7 August 2026; the ninth edition mentioned in the project notes was not supplied. Its §8 labels Theorem 1 internally audited but not externally refereed, and Theorem 2 conditional. Those are reported statuses, not proofs independently audited here. No cryptographic conclusion in this report depends on either theorem.

Correction to the project notes: direct evaluation gives 5 = 3 × 1 + 2, while 7 → 11 is non-decomposable. The eighth edition lists the exceptional primes as 2, 3, 7, rather than 2, 3, 5. [2]

02 / FACTORIZATION

An exact computational link

In ordinary two-prime RSA, the public modulus is N = pq. Knowing p and q allows computation of λ(N) = lcm(p − 1, q − 1), and hence the private exponent from the public exponent e. Factoring therefore suffices to recover the private key. This does not assert that every way of inverting RSA is equivalent to factoring. [3]

Consecutive integers expose the factoring subproblem

For the natural-number sequence, d = 1. At a = N + 1, the next term is b = N + 2, so l = N. Consequently, for every integer N > 1,

k(N + 1) = spf(N),    L(N + 1) = N/spf(N).

Here spf denotes the smallest prime factor. The proof is elementary: the least divisor greater than 1 must be prime, since a composite divisor would have a smaller divisor greater than 1. Thus a general-purpose WLJ oracle evaluated on (N + 1, N + 2) returns a factor of any composite N. Repeated calls factor the remaining cofactor. This is a direct consequence of the natural-number base case. [2]

Conditional algorithmic implication. If exact WLJ decomposition for arbitrary input pairs were computable in time polynomial in their bit length, integer factorization would be polynomial-time too. This is an implication about a hypothetical algorithm, not a claimed implementation. A routine specialized to consecutive primes would not automatically satisfy this premise.

A verified toy RSA example

The second row is the essential contrast in the mathematics: decomposition at N factors N − 1, whereas decomposition at N + 1 factors N. The arithmetic shift is simple; efficiently finding the required weight is the hard operation. In the second row, gcd(2, 10403) = 1.

Where the cost remains

The supplied PARI/GP kernel uses fordiv(l,k,...) to select the first admissible divisor. [1] Divisor enumeration can involve substantial factorization work; its compact syntax is not a complexity guarantee. Likewise, scanning up to √l costs exponentially many candidate steps in the bit length of l. A claimed speedup must count preprocessing, divisor discovery, and unsuccessful searches.

For any valid decomposition of a, gcd(a,k) = gcd(d,k), because a − d = kL. This exact identity explains when the weight can reveal a factor of a through a gcd. A nontrivial result requires 1 < gcd(d,k) < a; merely having a decomposition does not ensure it.

03 / RESIDUES & LEAKAGE

What WLJ could reveal about a key

Apply WLJ to a secret RSA prime p using its next prime, with coordinates (k_p, L_p, g_p). The exact modular statement is

p = k_pL_p + g_p,    p ≡ g_p (mod k_p).

Because 0 < k_p < p and p is prime, gcd(p,k_p) = gcd(g_p,k_p) = 1. If k_p and g_p become known, the public modulus N = pq therefore gives

q ≡ N g_p⁻¹ (mod k_p).

This is a genuine information channel under an explicit leakage assumption. In the checked example p = 101, q = 103, the pair (k_p,g_p) = (3,2) yields q ≡ 1 (mod 3). Such a small residue condition leaves many possible primes. Knowing the full triple reconstructs p immediately, which is simply disclosure of a different encoding of the secret.

The strongest exploratory route: partial information

Suppose independent information restricts L_p to L₀ + x with |x| small. Then a candidate factor takes the form p = k_p(L₀ + x) + g_p. This suggests comparing the resulting equation with partial-factor and small-root methods. Coppersmith established algorithms for sufficiently small roots under precise hypotheses, including applications to factoring with known leading bits. [5]

Research hypothesis. Particular leaked WLJ coordinates might furnish useful inputs to such methods. To establish this, one must specify what is public, how it is obtained, how small the unknown really is, and which theorem applies. Small g alone does not make the unknown L small. A congruence modulo an unknown divisor of N also differs from an ordinary known-modulus root problem; invoking “lattices” does not close that gap.

Three practical questions

The atlas describes sequence geometry. RSA normally publishes N and e, not the factors or their successor primes. Any proposed transfer from atlas patterns to a public-key attack must bridge that missing-information step.

04 / EXPLORATORY DIRECTIONS

Smoothness, generators, and limits

Shifted products and classical factoring methods

Pollard’s p − 1 method exploits prime factors p for which p − 1 divides a manageable exponent; smoothness makes such exponents feasible. Its standard gcd step uses a power modulo N. Elliptic-curve factoring similarly exploits favorable group-order factorization. [4] WLJ instead supplies p − g = kL. These shifted products invite comparison, but p − g is not generally the order of the multiplicative group modulo p.

A plausible project is to test whether WLJ features predict useful smoothness properties after conditioning on prime size and gap. Any value must exceed what direct partial factorization or residue tests already provide. Small weight alone is insufficient: it certifies one admissible divisor of p − g, while its cofactor L may still contain a large prime. No theorem here turns that information into smoothness of p − 1.

Prime-generation diagnostics

WLJ could provide descriptive features for comparing prime generators: weight columns, level classes, and gap-conditioned residues. The eighth edition’s random-sequence controls warn that aggregate rarefaction can track density effects; finer arithmetic differences require appropriately matched controls. [2, §§6–8] This motivates exploratory diagnostics, not a cryptographic randomness certificate.

A careful comparison would match bit length, sampling interval, parity, and small-prime exclusions. It would also distinguish independently sampled primes, next-prime searches from random starting points, and deliberately filtered families. Under uniform starting points in a fixed interval, next-prime mapping can assign larger probability to primes preceded by larger gaps, apart from endpoint effects. WLJ uses the forward gap, so the two effects must not be conflated.

NIST explicitly states that statistical testing cannot replace cryptanalysis, and its revision decision rejects using the SP 800-22 suite to assess cryptographic random-number generators. [6] A WLJ diagnostic deserves the same distinction between detecting a pattern and showing predictability.

Could WLJ underpin a new cryptographic primitive?

A deterministic coordinate transform creates no fresh entropy from a fixed input sequence. Publishing a full valid triple discloses a through kL + d. Hiding one coordinate can define a puzzle, but a cryptosystem additionally needs an efficient legitimate procedure, a precise adversary model, and a defensible hardness assumption. These are interesting design questions; the decomposition identity by itself supplies none of those security proofs.

The most promising constructive role is initially experimental: organize arithmetic instances, discover restricted families, and formulate a precise inversion problem. Whether that problem is new, useful, and resistant to existing methods remains to be established.

Beyond RSA

Elliptic-curve discrete logarithms concern a group operation, not the ordered successor structure of integers. ML-KEM is based on module-lattice problems. [7] No reduction from either problem to WLJ is established by the supplied materials or this analysis. A new connection would need to preserve the relevant algebra, rather than merely resemble a product-plus-remainder formula.

05 / RESEARCH AGENDA

A modest next step with a clear test

The most useful near-term goal is to determine whether WLJ exposes information that is both available and computationally useful. The following are proposed experiments, not results of this report.

Keep the first stage small and predeclare a time budget. Use the supplied PARI/GP kernel, retain failures as well as successes, and increase the workload only if a controlled result warrants it. No large census, cryptographic-size factorization, or attack benchmark was run for this report; numerical work was limited to the displayed examples and elementary boundary checks.

Conclusion

WLJ has an exact connection to factoring through its natural-number base case and an exact connection to residue leakage through p ≡ g (mod k). These support concrete questions about partial information, structured prime generation, and arithmetic diagnostics. The decisive future advance would be an efficient way to obtain useful coordinates, or a demonstrated predictive advantage after controlling for classical features. The evidence reviewed here establishes neither a general RSA attack nor a new secure primitive. It does provide a precise, testable route for exploration.

References and provenance

[1] Rémi Eismann, founding paper, arXiv:0711.0865; supplied decompwlj_fordiv.txt (canonical compact PARI/GP kernel). Definitions and base construction; later statuses taken from [2].

[2] Fable5_decompwlj_deep_analysis_8th_edition_final_2026_08_07.html, supplied project treatise, 7 August 2026. Consulted §§1, 3, 6–8 and the verification record. Internal research source; ninth edition unavailable. No census rerun or independent audit of Theorem 1.

[3] Moriarty et al., RFC 8017, PKCS #1 v2.2, especially §§3.1–3.2: RSA key structure.

[4] Menezes, van Oorschot & Vanstone, Handbook of Applied Cryptography, Chapter 3, §§3.2–3.3: factoring methods and the RSA problem.

[5] D. Coppersmith, Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities, J. Cryptology 10 (1997), 233–260.

[6] NIST, SP 800-22 Rev. 1a, and 19 April 2022 revision decision: limits of statistical testing.

[7] NIST, FIPS 203: ML-KEM, published 13 August 2024. External sources consulted 6 September 2026.