Silent-Radix Cryptography: Exploiting the Base Ambiguity of Positional Notation as a Cryptographic Primitive

In the paper's words

The silent radix — a positional numeral cannot internally specify its own base — is a cryptographic primitive. Silent Radix Encryption (SRE) uses a secret base b as a trapdoor for key encapsulation. Security reduces to an unknown-weight integer knapsack problem.

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

Numbers it reports

1. Hash pre-image resistance (128 bits for SHA-256 against quantum adversaries in Grover's algorithm) protects against this attack.

   from 3.3 Known-Plaintext Attacks

2. SREv2 KEM Unknown-weight knapsack 512–4096 decimal digits ✅ RSA-OAEP KEM Integer factorization 2048–4096 bits ❌ ECDH KEM Elliptic curve discrete log 256–521 bits ❌ Kyber (NIST PQC) KEM Module-LWE (lattice) 800–1568 bytes ✅ Classic McEliece KEM Goppa code decoding 261,120–1,357,824 bytes ✅ Advantages of SRE: No structured mathematics : unlike lattice or code-based schemes, SRE requires only elementary arithmetic (multiplication, addition, SHA-256).

   from 4. Comparison with Existing Schemes

3. Disadvantages of SRE: Large envelope size : for $\lambda = 128$, the envelope $N$ requires ~4096 decimal digits (~13,600 bits), larger than Kyber (1,568 bytes = 12,544 bits) but smaller than Classic McEliece (1.3 MB).

   from 4. Comparison with Existing Schemes

4. Therefore every base, when named using its own numeral system, calls itself "base-10." [established — §1.1 of the Ultrametric Foundation] The efficiency of positional notation is purchased at the permanent cost of latent ambiguity.

   from 1.1 The Silent Radix

5. For a secret base $b \sim 2^\lambda$: Security parameter $\lambda$ Candidate bases Brute-force cost Status 64 $\sim 2^{64}$ Feasible for large organizations Acceptable for prototype 128 $\sim 2^{128}$ Infeasible for classical computers Post-quantum resistant 256 $\sim 2^{256}$ Infeasible for any foreseeable computer Excessive margin The brute-force attack requires: for each candidate $b$, compute $K = \sum d_i \cdot b^i$ (an $O(k \log b)$ arithmetic operation), derive $k_{\text{sym}}$, decrypt $c$, and check for valid plaintext.

   from 3.2 Brute-Force Complexity

6. However, the polynomial has $k$ terms and degree $k-1$, which exceeds the range where Coppersmith's method is efficient for $k \gt 64$.

   from 3.4 Lattice Reduction

How it is done

1. Hash pre-image resistance (128 bits for SHA-256 against quantum adversaries in Grover's algorithm) protects against this attack.

   from 3.3 Known-Plaintext Attacks

2. For degree $k$, this is related to Coppersmith's method for finding small roots of polynomials modulo integers.

   from 3.4 Lattice Reduction

3. The signature is a proof of knowledge of $b$ such that $H(N \parallel m) = \text{commit}(b)$, where $H$ is a hash function and $\text{commit}(b) = \text{SHA-256}(\sum d_i \cdot b^i)$.

   from 5.1 Silent-Radix Signatures

4. Grover's algorithm provides a quadratic speedup for brute-force key search (from $2^{128}$ to $2^{64}$).

   from 6. Limitations and Open Problems

What would falsify it

The paper's text states no explicit falsification criterion; the limits below are what it says about where the claim stops.

Where the claim stops

1. The silent-radix KEM lacks a reduction to a standard cryptographic assumption.

   from 6. Limitations and Open Problems

2. The observation that a positional numeral cannot internally specify its own base — the "silent radix" problem identified in §1.1 of the Ultrametric Foundation — is not merely a philosophical curiosity.

   from Abstract

3. A numeral string cannot internally specify its own interpretation rule.

   from 1.1 The Silent Radix

Where to read

1. Exploiting the Base Ambiguity of Positional Notation as a Cryptographic Primitive143 words · 1 min
2. 1. Mathematical Foundation222 words · 1 min
3. 2. Silent Radix Encryption (SRE)391 words · 2 min
4. 3. Security Analysis342 words · 1 min
5. 4. Comparison with Existing Schemes201 words · 1 min
6. 5. Extensions and Variants197 words · 1 min
7. 6. Limitations and Open Problems151 words · 1 min
8. 7. Connection to the Ultrametric Foundation Thesis146 words · 1 min
9. 8. Conclusion98 words · 1 min
10. Changelog9 words · 1 min