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

Abstract

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. It is a cryptographic primitive. We formalize this as **Silent Radix Encryption (SRE)**, a key encapsulation mechanism (KEM) where the shared secret is derived from the base-$b$ interpretation of publicly transmitted decimal digits. Only a party who knows the secret base $b$ can recover the correct key. Eve, seeing the same decimal digit string $N$, must solve a variant of the integer knapsack problem where the weights ($b^i$) are unknown: given $N = \sum d_i \cdot 10^i$, find $b$ such that $K = \sum d_i \cdot b^i$ decodes the ciphertext correctly. For $b \sim 2^{128}$, brute force is infeasible. ---