The Silent Radix Is Not a Key: A Security Analysis of Unknown-Radix Encoding as a Cryptographic Primitive

In the paper's words

Positional numerals do not internally specify their own base, and a recent informal proposal — Silent Radix Encryption (SRE) — conjectures that this "silent radix" can serve as a cryptographic secret: plaintext symbols are mapped to digits by a secret map and read as an integer in a secret radix B, with B acting as a trapdoor for key encapsulation. No formal security analysis of this construction exists. We formalize the encoding core, define its key space over admissible radices B ∈ [2, 2¹6] and digit maps on a 26-symbol alphabet, and compute its entropy: $104.38$ bits.

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. The Multi-Factor Key Derivation Function (MFKDF) construction extends password-based key derivation with additional authentication factors such as TOTP, HOTP, and hardware tokens, and reports an exponential security improvement over PBKDFs at less than 12 ms of added latency [7].

   from 2. Background and Related Work

2. We formalize the encoding core, define its key space over admissible radices $B \in [2, 2^{16}]$ and digit maps on a 26-symbol alphabet, and compute its entropy: $104.38$ bits.

   from Abstract

3. We then show the construction reduces to a monoalphabetic substitution cipher with base-dependent segmentation, and we derive a two-stage ciphertext-only attack: a histogram test recovers $B$ at a cost of at most $6.5535 \times 10^{7}$ digit operations for a 1000-symbol ciphertext, after which standard frequency analysis applies within a unicity distance of $149.1$ symbols, computed from the key entropy and an assumed language redundancy of $0.70$ bits/symbol.

   from Abstract

4. A formal definition of the unknown-radix encoding scheme, its key space (admissible radices $B \in [2, B_{\max}]$ and digit maps over an $m$-symbol alphabet), and the computation of its entropy: $104.38$ bits for $m = 26$, $B_{\max} = 2^{16}$ (Sections 3 and 4.1).

   from 1. Introduction

5. A reduction showing that decryption under a known radix is exactly a monoalphabetic substitution with base-dependent segmentation, and a derivation of the unicity distance $n_0 = 149.1$ symbols for the full key space (Sections 4.2 and 4.3).

   from 1. Introduction

6. An explicit two-stage ciphertext-only attack with full arithmetic: radix recovery by histogram discrimination at a cost of at most $6.5535 \times 10^{7}$ digit operations for $n = 1000$, followed by frequency analysis of the residual substitution (Section 4.4).

   from 1. Introduction

How it is done

1. We then show the construction reduces to a monoalphabetic substitution cipher with base-dependent segmentation, and we derive a two-stage ciphertext-only attack: a histogram test recovers $B$ at a cost of at most $6.5535 \times 10^{7}$ digit operations for a 1000-symbol ciphertext, after which standard frequency analysis applies within a unicity distance of $149.1$ symbols, computed from the key entropy and an assumed language redundancy of $0.70$ bits/symbol.

   from Abstract

2. A recurring failure mode in cryptographic engineering is the deployment of primitives whose security rests on the secrecy of a design parameter rather than on the secrecy of a key under a public, well-scrutinized algorithm.

   from 1. Introduction

3. Kerckhoffs's principle — the system must remain secure even if everything about it except the key is public — is operationalized in modern formal methods: the first formal model of Bitcoin's Payment Protocol (BIP70) formalised the previously shown refund-attack vulnerability arising from lack of authentication of refund addresses [4].

   from 2. Background and Related Work

4. The key space is $$\mathcal{K} = \{(B, \pi) : B \in \{2, \ldots, B_{\max}\},\ \pi \in \mathrm{Sym}(m)\},$$ where $\mathrm{Sym}(m)$ is the set of bijections from $\Sigma$ to $\{0, \ldots, m-1\}$.

   from 3.2 Key space and adversary model

What would falsify it

1. We conclude that radix obscurity offers no security advantage over classical substitution ciphers and therefore no quantum-relevant one, and we state the conditions under which this negative result would be falsified.

   from Abstract

2. A statement of the negative result for quantum relevance — a scheme broken classically on a single ciphertext cannot offer a quantum-relevant advantage — together with falsification conditions and the limits of the analysis (Section 6).

   from 1. Introduction

3. Finally, work on cryptographically verifiable authorization for autonomous AI agents proposes a falsifiable hypothesis and proof of concept for giving agents' requests cryptographic evidence of policy compliance in context [1].

   from 2. Background and Related Work

Where the claim stops

1. The result is a negative one, and its strength depends entirely on the assumptions behind the arithmetic.

   from 6. Discussion

2. The result rests on the following assumptions, with the sensitivity of the headline numbers to each: Language model ($A_1$–$A_3$).

   from 6. Discussion

3. If the full SRE construction adds layers that randomize digit frequencies or re-encrypt per message, the Stage 1 histogram test may fail; the source proposal's supplied description states only the trapdoor claim and the knapsack reduction, and gives no such layer, so we cannot assess one.

   from 6. Discussion

Where to read

1. Abstract225 words · 1 min
2. 1. Introduction599 words · 3 min
3. 2. Background and Related Work738 words · 3 min
4. 3. Methods560 words · 2 min
5. 4. Analysis1,352 words · 6 min
6. 5. Results248 words · 1 min
7. 6. Discussion571 words · 2 min
8. 7. Practical Implications and Engineering Pathways622 words · 3 min
9. 8. Conclusion215 words · 1 min
10. Provenance and AI Involvement197 words · 1 min