Algorithmically Presented Numbers and Canonical Representations in Cryptographic Protocols

• URL: http://arxiv.org/abs/2607.05016v1
• Date: Mon, 06 Jul 2026 12:56:58 GMT
• Title: Algorithmically Presented Numbers and Canonical Representations in Cryptographic Protocols
• Abstract summary: The focus is not solely on the computability of the abstract value as an extensional property, but on the algorithmic structure of its presentation in a representation system.<n>We distinguish three representation-theoretic notions -- algorithmically approximable (A_app), the computable real numbers, and finitely exactly describable in a system.<n>We show that there is no computable extensional canonicalizer that uniformly transforms arbitrary approximation programs of computable real numbers into unique finite value encodings.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: This paper develops a representation-theoretic perspective on cryptographic protocols. The focus is not solely on the computability of the abstract value as an extensional property, but on the algorithmic structure of its presentation in a representation system: for operational use in protocols, algorithmic accessibility of the value does not suffice; its fixed presentation is also decisive. We distinguish three representation-theoretic notions -- algorithmically approximable (A_app, the computable real numbers), finitely exactly describable in a system (A_fin(S)), and canonical normalizability of a system -- and show that there is no computable extensional canonicalizer that uniformly transforms arbitrary approximation programs of computable real numbers into unique finite value encodings. As the operational rational core presentation we use the rational system with its canonical encoding specification Sigma_Q (fixed rules for valid fraction descriptions, canonical codes, and normalization); the associated value set is A_ex = Q. The notion of a canonically serializable object class transfers this core idea to practical protocol objects (files as byte sequences, hash values, transaction IDs, and normatively serialized payloads). We illustrate the consequences for interoperability, well-definedness, and verification with fully worked toy examples from symmetric and asymmetric encryption and hashing, and with a real-world example, the snaproot hash-anchoring protocol for blockchain-based file integrity verification. The paper thereby shows that the mathematical determinacy of a value and its operational uniqueness as a protocol object are two different requirements. Once a normative representation specification has been fixed, byte-level correctness and well-definedness arguments can be carried out without further implementation-dependent serialization or rounding decisions.

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