[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-83881-en":3,"doc-seo-83881-105":30,"detail-sidebar-cat-0-en-105":84},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":21,"is_downloadable":21,"audit_status":21,"page_count":22,"language":23,"language_code":24,"site_id":25,"html_lang":24,"table_of_contents":26,"faqs":27,"seo_title":13,"seo_description":14,"update_tm":28,"read_time":29},83881,8796095461564,"Liam","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Algorithmically Presented Numbers and Canonical Representations in Cryptographic Protocols","The paper develops a representation-theoretic perspective on cryptographic protocols, separating the extensional determinacy of real values from the algorithmic structure of their operational presentation. It distinguishes algorithmic approximability, finite exact describability in a system, and canonical normalizability, and proves the absence of a computable extensional canonicalizer that uniformly converts arbitrary approximation programs into unique finite encodings. Using a rational system with canonical encoding ΣQ, it applies canonically serializable object classes to protocol artifacts and studies interoperability, well-definedness, and verification via encryption and hashing examples, including snaproot hash-anchoring for blockchain file integrity verification.","arXiv :2607 .050 16v 1 [ cs .CR] 6 Jul 2026  \nAlgorithmically Presented Numbers and Canonical Representations in Cryptographic Protocols  \nArslan Brömme 1  \nChain Horizon GmbH  \nVersion [0.9.5.20](0.9.5.20) (Working Draft)  \nPreprint / working paper. This version is a work in progress and may be updated. Comments are welcome . This document is an English translation of the corresponding  \nGerman version (v [0.9.5.17](0.9.5.17)) of this paper.  \nAbstract  \nThis paper develops a representation-theoretic perspective on cryptographic protocols. The focus isnot 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; in addition, its fixed presentation is 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 Σ Q (the fixed rules for valid fraction descriptions, canonical codes, and normalization); the associated value set is Aex = Q. The notion of a canonically serializable object class transfers this core idea to practical protocol objects (concrete 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-out toy examples from symmetric and asymmetric encryption as well as hashing, and with a real-world application 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.  \nKeywords— computable real numbers, canonical serialization, representation theory, cryptographic protocols, canonicalization barrier, computable analysis, protocol interoperability  \n1 Introduction and Motivation  \nReal numbers are central objects of analysis and of many mathematical models. In cryptography and theoretical computer science, however, there is a fundamental tension between mathematical uniqueness and algorithmic reproducibility: real numbers are uniquely defined as mathematical objects, but in general possess no finite, canonical representation. Cryptographic protocols, by contrast, process finite byte sequences. If these byte sequences are understood as representations of semantic objects, then a uniquely specified encoding is helpful for representation-invariant processing. If semantically equal objects are to produce identical protocol bytes independently of their initial representation, a canonical encoding or normalization is additionally required; we call this stronger requirement byte-input invariance.  \nSince this paper connects notions from computable analysis and from cryptography, we first fix some basic terminology. A cryptographic primitive is a basic building block such as a hash function, an encryption scheme, or a signature scheme. By a protocol we mean a normative specification of inputs,  \n1 Dipl.-Inform. , B.Sc. , CISSP, CISA, CISM, [CAISE.](CAISE. arslanb@chain-horizon.com)[ arslanb@chain-horizon.com](CAISE. arslanb@chain-horizon.com)  \nAlgorithmically Presented Numbers and Canonical Representations in Cryptographic Protocols 1  \nrepresen","cbCaig1dmJ9LcJRY","https://ap.wps.com/l/cbCaig1dmJ9LcJRY","pdf",525618,5,1,22,"English","en",105,"# Abstract\n# Introduction and Motivation\n## Motivation: uniqueness vs reproducibility\n## Core terminology and definitions\n## Representation-invariant and byte-input invariance\n# Representation-theoretic notions in cryptographic contexts","[{\"question\":\"Does the paper claim a universal computable method for canonicalizing arbitrary real-number approximations?\",\"answer\":\"No. It shows there is no computable extensional canonicalizer that uniformly transforms arbitrary approximation programs of computable real numbers into unique finite encodings.\"}]",1784191195,55,{"code":4,"msg":31,"data":32},"ok",{"site_id":25,"language":24,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":79,"head_meta":81,"extra_data":83,"updated_unix":28},"algorithmically-presented-numbers-and-canonical-representations-in-cryptographic-protocols","",{"@graph":36,"@context":78},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":21},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/algorithmically-presented-numbers-and-canonical-representations-in-cryptographic-protocols/83881/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":24,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-07-25","2026-07-16",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72],{"name":73,"@type":74,"acceptedAnswer":75},"Does the paper claim a universal computable method for canonicalizing arbitrary real-number approximations?","Question",{"text":76,"@type":77},"No. It shows there is no computable extensional canonicalizer that uniformly transforms arbitrary approximation programs of computable real numbers into unique finite encodings.","Answer","https://schema.org",{"og:url":52,"og:type":80,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":82,"canonical":52},"index,follow",{"doc_id":7,"site_id":25},{"code":4,"msg":5,"data":85},[86,90,94,98,102,107,112,115,120,123,127],{"id":21,"doc_module":4,"doc_module_name":46,"category_name":87,"show_sort_weight":88,"slug":89},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":91,"show_sort_weight":92,"slug":93},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Exam",70,"exam",{"id":20,"doc_module":4,"doc_module_name":46,"category_name":99,"show_sort_weight":100,"slug":101},"Comic",60,"comic",{"id":103,"doc_module":4,"doc_module_name":46,"category_name":104,"show_sort_weight":105,"slug":106},6,"Technology",50,"technology",{"id":108,"doc_module":4,"doc_module_name":46,"category_name":109,"show_sort_weight":110,"slug":111},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":113,"slug":114},30,"research-report",{"id":116,"doc_module":4,"doc_module_name":46,"category_name":117,"show_sort_weight":118,"slug":119},9,"Religion & Spirituality",20,"religion-spirituality",{"id":118,"doc_module":4,"doc_module_name":46,"category_name":121,"show_sort_weight":118,"slug":122},"World Cup","world-cup",{"id":124,"doc_module":4,"doc_module_name":46,"category_name":125,"show_sort_weight":124,"slug":126},10,"Lifestyle","lifestyle",{"id":128,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":20,"slug":130},19,"General","general"]