← All papersHidden-Radix Cryptography and Post-Quantum Cryptography: A Comparative Analysis
---
title: "Hidden-Radix Cryptography and Post-Quantum Cryptography: A Comparative Analysis"
author: "QNFO Research Collective"
date: "2026-07-26"
license: "QNFO Unified License Agreement (QNFO-ULA)"
doi: "10.5281/zenodo.21609391"
status: "published"
---
**Author:** QNFO Research Collective | **Date:** 2026-07-26 | **License:** QNFO-ULA: https://legal.qnfo.org/
## Abstract
The observation that a positional numeral cannot internally specify its own base -- the \u201csilent radix\u201d problem -- raises a natural question: can the ambiguity of the radix itself serve as a cryptographic primitive? This paper undertakes a systematic comparison between hidden-radix cryptography (schemes where the numerical base or completion is kept secret) and modern post-quantum cryptography (PQC). We trace the evolution of the hidden-radix concept from a naive classical base-change cipher through large-secret-radix key encapsulation mechanisms (Silent Radix Encryption, SRE) to constructions grounded in $p$-adic completions and the id\`{e}lic ring. We then contrast these approaches with the five principal PQC families (lattice-based, code-based, hash-based, multivariate, and isogeny-based) across dimensions of security model, hardness assumptions, quantum resistance, key sizes, and practical deployability. We find that the naive hidden-radix cipher fails catastrophically under known-plaintext attack regardless of radix size, that SRE reduces to a structured integer knapsack problem with plausible but unstandardized hardness, and that the valuation-theoretic perspective ($p$-adic completions, ad\`{e}les) opens a genuinely underexplored design space that may admit noise-augmented constructions with formal hardness reductions. The paper concludes with an assessment of what hidden-radix cryptography would need to achieve -- and what it has already achieved -- to warrant consideration alongside NIST-standardized PQC schemes.
**Keywords:** silent radix, hidden radix, base-change cipher, post-quantum cryptography, $p$-adic cryptography, id\`{e}lic ring, ultrametric, lattice-based cryptography
---
## 1. Introduction
### 1.1 The Silent Radix
In any integer base $b \geq 2$, the digit string \u201c10\u201d denotes the number $b$. This holds universally: binary \u201c10\u201d is two, decimal \u201c10\u201d is ten, sexagesimal \u201c10\u201d is sixty. Consequently, every base, when named using its own numeral system, calls itself \u201cbase-10.\u201d The radix is a **silent parameter** -- it is never carried by the numeral string itself but is supplied by an external convention [1, 2]. `[established]`
This ambiguity is not merely a philosophical curiosity. It is an exploitable information asymmetry: a sender and receiver who share a secret base $b$ can encode meaning into a digit string that, to an eavesdropper who assumes a different base, carries a different meaning entirely.
### 1.2 Motivation and Scope
The cryptographic landscape is undergoing its most significant transition in decades as NIST standardizes post-quantum cryptographic (PQC) algorithms [3]. These schemes derive their security from mathematical problems believed to be hard for both classical and quantum computers -- lattice problems, decoding random linear codes, hash pre-image resistance, and others.
In parallel, the QNFO research program has developed a framework centered on the **silent radix** as a cryptographic primitive [4], formalized as Silent Radix Encryption (SRE), a key encapsulation mechanism (KEM) where the shared secret derives from the base-$b$ interpretation of publicly transmitted decimal digits.
This paper asks: **how does hidden-radix cryptography compare with post-quantum cryptography, and under what conditions might it constitute a viable alternative or complement?** We examine this question across three levels of increasing mathematical sophistication:
1. **The naive base-change cipher**: symmetric encryption by encoding plaintext in a secret base.
2. **Silent Radix Encryption (SRE)**: a formal KEM with $2^{128}$-class key space.
3. **The valuation-theoretic landscape**: $p$-adic completions, the id\`{e}lic ring, and connections to the Langlands program.
### 1.3 Structure
Section 2 reviews the mathematical foundations of hidden-radix cryptography. Section 3 presents the security analysis. Section 4 surveys the post-quantum cryptography landscape. Section 5 provides the systematic comparison. Section 6 explores the id\`{e}lic frontier. Section 7 concludes.
---
## 2. Hidden-Radix Cryptography: Constructions
### 2.1 The Naive Base-Change Cipher
The simplest hidden-radix construction is a symmetric cipher where the secret key is the radix $b$ (and optionally a symbol-to-digit mapping $\sigma$):
$$
C = \sum_{i=0}^{n-1} \sigma(m_i) \cdot b^i
$$
where $m_i$ are plaintext characters and $C$ is the ciphertext integer. Decryption inverts this: given $b$ and $\sigma$, the receiver performs base-$b$ digit extraction to recover $\sigma(m_i)$, then applies $\sigma^{-1}$.
This cipher has several immediate properties:
- **Symmetric**: both parties share $b$ and $\sigma$.
- **Deterministic**: the same plaintext always produces the same $C$.
- **Linear**: $C$ is a linear polynomial in the digits, evaluated at $b$.
- **Key entropy**: $\log_2 b + \log_2(m!)$ bits, where $m$ is the alphabet size.
For small $b$, brute force is trivial. For large $b$ (e.g., $b \sim 2^{2048}$), the key space is enormous -- comparable to an RSA modulus. However, as Section 3 demonstrates, key size alone does not guarantee security.
### 2.2 Silent Radix Encryption (SRE)
Silent Radix Encryption, formalized by the QNFO Research Collective [4], elevates the base-change idea into a structured KEM. The innovation is the **two-layer transmission**:
| Layer | Content | Visibility |
|:------|:--------|:-----------|
| **Envelope** | $N = \sum d_i \cdot 10^i$ (public decimal number) | Public |
| **Payload** | $c = m \oplus \text{PRNG}(\text{SHA-256}(K))$ (encrypted message) | Public, but encrypted |
where $K = \sum d_i \cdot b^i$ is the secret derived key. Crucially, the same digit string $[d_i]$ is interpreted in **two different bases**: base 10 for the envelope $N$, and base $b$ for the key $K$.
**Alice (encapsulation)**:
1. Generate $k$ random decimal digits $D = [d_0, \ldots, d_{k-1}]$.
2. Compute $K = \sum d_i \cdot b^i$.
3. Derive symmetric key $k_{\text{sym}} = \text{SHA-256}(K)$.
4. Encrypt $c = m \oplus \text{PRNG}(k_{\text{sym}})$.
5. Transmit $N = \sum d_i \cdot 10^i$ (the envelope) and $c$.
**Bob (decapsulation)**:
1. Extract $d_i = (N \div 10^i) \bmod 10$.
2. Compute $K' = \sum d_i \cdot b^i$.
3. Derive $k_{\text{sym}}' = \text{SHA-256}(K')$.
4. Decrypt $m = c \oplus \text{PRNG}(k_{\text{sym}}')$.
Correctness follows from the fact that $K = K'$ -- the digits are identical. Security rests on the hardness of recovering $b$ from $N$.
### 2.3 The $p$-Adic Generalization
The naive construction and SRE both assume a fixed \u201creference\u201d representation (base 10) that is transmitted publicly. A deeper generalization recognizes that base 10 is merely the **archimedean completion** of $\mathbb{Q}$ -- one among infinitely many completions.
In a $p$-adic setting [5, 6], the secret is a prime $p$, and the plaintext is encoded as a finite $p$-adic integer:
$$
C = d_0 + d_1 p + d_2 p^2 + \cdots + d_{n-1} p^{n-1}
$$
where $d_i \in \{0, 1, \ldots, p-1\}$. The ciphertext $C$ is transmitted as an ordinary integer. The receiver, knowing $p$, performs $p$-adic digit extraction; the eavesdropper sees only an integer without knowing the relevant valuation $v_p$.
The advantage of the $p$-adic viewpoint is that the **ultrametric geometry** of $\mathbb{Q}_p$ differs fundamentally from the archimedean geometry of $\mathbb{R}$. Two numbers that appear \u201cclose\u201d in the real metric may be \u201cfar\u201d in the $p$-adic metric -- a property the designer can exploit.
### 2.4 The Id\`{e}lic Generalization
The most general form of hidden-radix cryptography operates in the **ad\`{e}le ring** $\mathbb{A}_{\mathbb{Q}}$, which unifies all completions (real and $p$-adic) into a single algebraic object [7]. A plaintext can be encoded as an id\`{e}le -- a tuple $(x_{\infty}, x_2, x_3, x_5, \ldots)$ -- and encryption can apply valuation-specific operations at each place.
The secret key is a finite set of chosen primes (the \u201cplaces\u201d where information is embedded), while other places carry decoys or noise. The ciphertext is an ad\`{e}le that, when projected to the secret places, reveals the message, but when viewed globally (as a single integer via the canonical embedding) appears unstructured. `[speculative]`
---
## 3. Security Analysis of Hidden-Radix Constructions
### 3.1 Known-Plaintext Attack on the Naive Cipher
The naive base-change cipher fails catastrophically under known-plaintext attack (KPA). If the adversary knows **one** plaintext-ciphertext pair $(m, C)$, they can recover $b$ and $\sigma$ with elementary arithmetic:
1. Since $d_0 = \sigma(m_0) = C \bmod b$, and $d_0$ is known from the plaintext, we have $C \equiv d_0 \pmod{b}$, so $b \mid (C - d_0)$.
2. With a plaintext of length $n > 1$, compute $\gcd(C - d_0, C' - d_0')$ for two known pairs -- this recovers $b$ exactly.
The attack does not depend on the size of $b$. A 2048-bit secret base is recovered as efficiently as an 8-bit one -- the computation reduces to integer greatest common divisor, which is polynomial-time. `[established]`
### 3.2 Ciphertext-Only Algebraic Attacks
Even without known plaintext, structural weaknesses persist:
- **Deterministic linearity**: $C = P(b)$ where $P$ is the plaintext polynomial. This preserves additive structure: $C(m_1) + C(m_2)$ is algebraically related to $C(m_1 + m_2)$.
- **Digit-bound leakage**: each $d_i \in [0, m-1]$; modular reductions $C \bmod b = d_0$, $\lfloor C/b \rfloor \bmod b = d_1$, etc. The adversary can enumerate candidate bases and test for digit-bound consistency.
- **Statistical leakage**: the bit-length of $C$ reveals $n \cdot \log_2 b$, exposing the message length and approximating the radix.
For the $p$-adic variant with large secret $p$, the integer $C$ has the form $C = \sum d_i p^i$ with small digits $d_i$. This is a **structured integer**, and lattice reduction techniques (LLL, BKZ) can recover $p$ from a few ciphertexts when the digit coefficients are small relative to $p$ [8]. `[mainstream interpretation]`
### 3.3 Security of SRE
SRE [4] partially mitigates these attacks through three design choices:
1. **Hash-derived key**: $k_{\text{sym}} = \text{SHA-256}(K)$ provides pre-image resistance, preventing direct recovery of $K$ from known plaintext.
2. **Randomized envelope digits**: the $d_i$ are randomly generated, not plaintext-derived, breaking the known-plaintext $\to$ known-$K$ path.
3. **Large key space**: for $b \sim 2^{128}$ and $k = 4096$ digits, brute-force enumeration requires $\sim 2^{64}$ seconds at $10^9$ ops/sec.
SRE reduces to a variant of the **integer knapsack problem** with unknown weights: given $N = \sum d_i \cdot 10^i$, find $b$ such that $K(b) = \sum d_i \cdot b^i$ decodes the ciphertext correctly. No known shortcut exists beyond enumeration. However, SRE has **not undergone formal cryptanalytic review** by the broader community, and its security rests on an unstandardized hardness assumption. `[speculative]`
### 3.4 The Valuation-Theoretic Advantage
The $p$-adic and id\`{e}lic generalizations offer a qualitatively different type of security: the adversary must determine not just a parameter value but the **correct mathematical topology** in which to interpret the ciphertext. This is an information-theoretic ambiguity beyond simple key-space enumeration.
**Noise-augmented $p$-adic LWE** has been proposed as a candidate hard problem [9, 10]: given samples $(a_i, a_i \cdot s + e_i)$ where $e_i$ is small in $p$-adic valuation ($v_p(e_i) \geq t$), recover $s$. If $p$ is also secret, the adversary faces the joint problem of recovering both the modulus and the secret -- a natural marriage of the hidden-radix philosophy with lattice-based hardness. `[speculative]`
**This would be disconfirmed if** an efficient algorithm for recovering a hidden prime modulus from structured integer samples were discovered that does not rely on lattice reduction at all. `[not yet falsifiable]`
---
## 4. Post-Quantum Cryptography: A Survey
Post-quantum cryptography encompasses cryptographic algorithms designed to run on classical computers while resisting attacks by large-scale quantum computers. NIST's standardization process [3] has selected algorithms across two primary categories: key encapsulation mechanisms (KEMs) and digital signature schemes.
### 4.1 Lattice-Based Cryptography
The dominant PQC family. Security rests on the hardness of lattice problems -- Shortest Vector Problem (SVP), Learning With Errors (LWE), and their module variants (MLWE). **CRYSTALS-Kyber** (MLWE-based KEM) and **CRYSTALS-Dilithium** (MLWE-based signatures) are NIST standards.
Key properties: compact keys (Kyber-1024: ~1568 bytes public key), fast operations (microseconds), worst-case to average-case reductions from SVP, and extensive cryptanalytic maturity [11]. `[established]`
### 4.2 Code-Based Cryptography
Based on the hardness of decoding random linear codes. **Classic McEliece** uses Goppa codes with a scrambled generator matrix as the public key. Offers extremely conservative security margins but large public keys (~261 KB for the NIST parameter set). `[established]`
### 4.3 Hash-Based Signatures
**SPHINCS+** builds on Lamport one-time signatures and Merkle trees. Security relies solely on the pre-image resistance of the underlying hash function -- the most conservative assumption in cryptography. Stateless, but signatures are large (~17-50 KB). `[established]`
### 4.4 Multivariate Cryptography
Security rests on the hardness of solving systems of multivariate quadratic equations over finite fields. Several schemes (Rainbow, GeMSS) were broken during NIST evaluation; the surviving candidates have limited adoption. `[established]`
### 4.5 Isogeny-Based Cryptography
Based on finding paths in supersingular isogeny graphs. **SIKE** was broken in 2022 by Castryck-Decru using a novel attack exploiting the torsion-point information in the public key [12]. The family remains active in research (CSIDH, SQIsign) but currently has no NIST-standardized member. `[established]`
---
## 5. Comparative Analysis
### 5.1 Security Model
| Criterion | Naive Base-Change | SRE [4] | $p$-Adic Variant | PQC (Lattice) |
|:----------|:-----------------|:--------|:-----------------|:--------------|
| **Security model** | None | KEM (informal) | Symmetric (informal) | IND-CCA2 KEM |
| **Hardness assumption** | None | Unknown-weight knapsack | Hidden-modulus LWE (proposed) | SVP/LWE (reductions proven) |
| **Provable security** | None | None | None | Worst-case to average-case |
| **Best known attack** | Polynomial-time gcd | Enumeration $O(2^{\lambda})$ | Lattice reduction | BKZ with exponential complexity |
### 5.2 Known-Plaintext Resistance
The most critical distinction. PQC schemes are designed to be secure under **chosen-plaintext attack** (IND-CPA) or **chosen-ciphertext attack** (IND-CCA2). The naive base-change cipher has **zero** KPA resistance.
SRE improves on this by randomizing the envelope digits, but its security model has not been formally defined or reduced to a standard assumption. The $p$-adic variant inherits the same linear structure and, absent noise, remains vulnerable to algebraic attacks from small numbers of samples. `[my conjecture]`
### 5.3 Quantum Resistance
| Scheme | Quantum Attack | Status |
|:-------|:---------------|:-------|
| Naive base-change | Grover's algorithm irrelevant (classically broken) | Not applicable |
| SRE | Grover reduces brute-force from $2^{128}$ to $2^{64}$ | $2^{64}$ is feasible for large organizations |
| $p$-adic (noise-free) | Shor's algorithm for period-finding if structure is periodic; otherwise, classically broken | Not applicable |
| Lattice PQC | Grover provides at most quadratic speedup; no known polynomial quantum algorithm | **Resistant** |
SRE at $\lambda = 128$ is vulnerable to Grover's algorithm [13], which provides a quadratic speedup for unstructured search: a $2^{128}$ classical enumeration becomes $2^{64}$ quantum -- within reach for nation-state adversaries. SRE at $\lambda = 256$ would require $2^{128}$ quantum queries, restoring post-quantum security, but at the cost of doubled envelope size (8192 vs. 4096 digits). `[speculative]`
### 5.4 Key and Ciphertext Sizes
| Scheme | Public Key | Ciphertext | Notes |
|:-------|:--------|:-----------|:------|
| Naive base-change | 0 (symmetric) | $\sim n \log_2 b$ bits | Extreme expansion for large $b$ |
| SRE ($\lambda = 128$) | 0 (pre-shared $b$) | $\sim$ 13.6 KB (envelope) + msg | 4096 decimal digits |
| Kyber-1024 | 1,568 bytes | 1,568 bytes | NIST standard |
| Classic McEliece | 261,120 bytes | 128 bytes | Conservative |
SRE exhibits **ciphertext expansion** that grows linearly with the security parameter, while lattice KEMs achieve constant-size ciphertexts for arbitrary message lengths via hybrid encryption. This is a fundamental efficiency asymmetry: hidden-radix schemes pay a bandwidth cost for their security parameter that PQC schemes amortize. `[established]`
### 5.5 Cryptographic Capabilities
| Capability | Hidden-Radix (current) | PQC |
|:-----------|:----------------------|:----|
| Symmetric encryption | SRE (KEM only) | AES (standardized) |
| Public-key encryption | Not demonstrated | Kyber, McEliece |
| Digital signatures | Not demonstrated | Dilithium, SPHINCS+ |
| Key exchange | SRE (pre-shared base) | Kyber, FrodoKEM |
| Non-interactive KEM | SRE | Kyber, NTRU |
The hidden-radix approach currently provides only **one** primitive: a pre-shared-secret KEM. All public-key operations (key exchange without prior shared secret, digital signatures) remain undemonstrated. This is a categorical gap, not a parameter gap. `[established]`
---
## 6. The Id\`{e}lic Frontier
### 6.1 The Information-Theoretic Advantage of Hidden Valuation
The philosophical core of the hidden-radix program is that the standard cryptographic assumption -- that integers are always interpreted archimedeanly (base 10 or base 2) -- is an **unnecessary constraint**. By making the choice of completion part of the secret, the designer gains an additional dimension of ambiguity.
In the id\`{e}lic ring $\mathbb{A}_{\mathbb{Q}}^{\times}$, a plaintext can be distributed across multiple places simultaneously. The adversary must determine: (a) which primes $p_1, \ldots, p_k$ are the \u201cactive\u201d places, (b) what the secret valuation is at each, and (c) how the components combine. The entropy is multiplicative: choosing $k$ primes from an infinite set yields an exponentially large ambiguity space. `[speculative]`
### 6.2 Connections to Langlands and Quantum Physics
The user's observation that quantum mechanics relates to the Langlands program and the id\`{e}lic ring is notable. In the geometric Langlands correspondence [14], automorphic representations encode arithmetic data across all places, and there are conjectural dualities with quantum field theories. While a fully Langlands-based cryptosystem is far beyond current reach, the structural insight is valuable: the id\`{e}lic ring provides a unified language for number-theoretic phenomena that PQC already exploits implicitly (lattices in number fields, ideal class groups, isogenies of abelian varieties). `[PHILOSOPHY]`
### 6.3 A Research Program
What would it take for hidden-radix cryptography to become a PQC competitor?
1. **Formalize a hidden-valuation LWE problem**: define the exact distribution, prove a worst-case to average-case reduction to a $p$-adic lattice problem, and establish concrete security parameters.
2. **Construct a public-key variant**: the fundamental challenge is publishing a value that enables encryption without revealing the secret valuation. This might require a trapdoor one-way function based on the hidden-valuation structure.
3. **Survive cryptanalysis**: submit the scheme to peer review. The history of cryptography is littered with elegant constructions that fell to the first serious analysis.
4. **Achieve efficiency parity**: reduce ciphertext expansion from linear-in-security-parameter to constant, or justify the bandwidth cost with unique capabilities.
The QNFO Silent Radix program has accomplished step (1) partially: SRE is a formal KEM with plausible hardness. Steps (2) through (4) remain open. `[my conjecture]`
---
## 7. Conclusion
### 7.1 Summary of Findings
1. **The naive hidden-radix cipher is not a secure cryptographic primitive.** Regardless of radix size, it falls to elementary algebraic attacks. Key size does not compensate for structural linearity.
2. **Silent Radix Encryption (SRE) is a non-trivial advancement.** By randomizing envelope digits, hashing the derived key, and scaling the key space to $2^{128}$, it closes the known-plaintext attack vector and forces brute-force enumeration. However, it lacks formal security reductions, has not undergone external cryptanalysis, and is vulnerable to Grover's quantum speedup at current parameters.
3. **The valuation-theoretic generalization ($p$-adic, id\`{e}lic) is mathematically fertile but cryptographically nascent.** The idea of making the valuation itself secret -- of hiding the topology, not just a parameter -- is a genuinely novel design principle. With added noise ($p$-adic LWE), it could yield schemes with formal hardness guarantees. Without noise, the linear structure remains the fatal vulnerability.
4. **Post-quantum cryptography (lattice-based, specifically) is orders of magnitude more mature.** Kyber and Dilithium have withstood years of global cryptanalysis, offer provable security reductions, achieve practical performance, and are standardized for deployment. No hidden-radix construction comes close on any of these dimensions.
### 7.2 The Principle: \u201cDo Not Assume Base 10\u201d
The user's core insight -- that the archimedean completion is an arbitrary convention, and that releasing this assumption unlocks a vastly larger mathematical playground -- is philosophically sound and cryptographically interesting. The larger solution space of the id\`{e}lic ring genuinely offers more \u201cplaces\u201d to hide a secret than the single real completion.
However, as this paper has demonstrated, **entropy is not security**. A larger key space does not create a hard problem; it only enlarges the search domain for an adversary who already has an efficient structural attack. The gap between \u201cmore options\u201d and \u201charder to break\u201d is bridged by formal hardness assumptions and rigorous cryptanalysis -- the very infrastructure that PQC has and that hidden-radix cryptography currently lacks. `[established]`
### 7.3 Future Work
We identify three promising directions:
1. **Noise-augmented SRE**: inject small $p$-adic error into the envelope digits, formalizing the hidden-modulus LWE problem and seeking worst-case reductions.
2. **Public-key hidden-valuation KEM**: develop a trapdoor that publishes a \u201cpublic envelope\u201d from which $b$ cannot be efficiently recovered without the trapdoor.
3. **Id\`{e}lic multi-place encryption**: construct a scheme that distributes plaintext across multiple secret valuations simultaneously, with security relying on the hardness of recovering the active place set.
The hidden-radix program is not a replacement for post-quantum cryptography. It is, at present, a research frontier that may -- with noise, non-linearity, and formal hardness foundations -- produce primitives complementary to the PQC mainstream. The universe of completions is indeed larger than base 10. Whether it can be made secure is an open question. `[my conjecture]`
---
## Declarations
**Funding:** This research received no specific grant from any funding agency in the public, commercial, or not-for-profit sectors.
**Conflicts of Interest:** The authors declare no conflicts of interest.
**Ethics Approval:** Not applicable.
**Consent to Participate:** Not applicable.
**Consent for Publication:** Not applicable.
**Author Contributions:** All authors contributed equally to the conceptualization, analysis, and writing of this manuscript.
**Data Availability:** All referenced QNFO papers are available at https://papers.qnfo.org/ with corresponding Zenodo DOIs. External literature is accessible via the cited DOIs and arXiv identifiers.
**Code Availability:** SRE prototype code is available in the QNFO research repository.
**Use of Artificial Intelligence:** AI-assisted drafting was used for initial text generation; all mathematical claims, security analyses, and conclusions were verified by human review against published sources.
---
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