{"schema":"https://ver.cy/schemas/card/1.0.0","id":"vr.tr.primitive-root-modulo-n","code":"thing-q948010","url":"https://ver.cy/models/thing/q948010/","name":"primitive root modulo n","alternateNames":[],"kind":"thing","status":"published","version":"0.2.0-wave.3","language":"en","classifiers":{"family":"Thing Registry","category":"Cross-cutting context","entryKind":"thing","plane":"XCT","domain":["XCT.QTY"],"industry":[],"navPath":"","tags":[],"facets":{}},"whatItIs":"In number theory, an integer g such that every integer coprime to n is congruent to some power of g modulo n, that is, g generates the multiplicative group of integers modulo n; primitive roots exist exactly when n is 1, 2, 4, a power of an odd prime, or twice a power of an odd prime, and they underlie discrete logarithms and several cryptographic protocols.","purpose":"Let an agent define primitive roots modulo n precisely, state when they exist and how many there are, test and find them for a given modulus, and relate them to orders, discrete logarithms and cryptography without giving operational key material.","scope":{"in":[],"out":[],"boundaries":[]},"distinguishingFeatures":["Generator of the multiplicative group modulo n","Existence depends on the form of n","Order equals Euler phi of n","Basis of the discrete logarithm"],"structure":{"bundles":[{"id":"define","name":"Define","description":"The definition.","layers":[{"id":"definition","name":"Definition","description":"Definition.","findings":[{"id":"definition-finding","name":"Definition","description":"Definition.","questions":[{"text":"What is a primitive root modulo n, and how does it relate to multiplicative order?","kind":"definition"},{"text":"Is the question about primitive roots modulo n or about complex roots of unity?","kind":"boundary"}]}]},{"id":"existence","name":"Existence","description":"Existence theorem.","findings":[{"id":"existence-finding","name":"Existence","description":"Existence.","questions":[{"text":"For which n do primitive roots exist, and how many are there?","kind":"definition"},{"text":"Which entry fits the multiplicative group modulo n?","kind":"action"}]}]}]},{"id":"compute","name":"Compute","description":"Finding and testing.","layers":[{"id":"test","name":"Test","description":"Testing.","findings":[{"id":"test-finding","name":"Test","description":"Test.","questions":[{"text":"How is an integer tested as a primitive root using the prime factors of phi of n?","kind":"action"},{"text":"What is the result for the modulus in question?","kind":"measurement"}]}]},{"id":"find","name":"Find","description":"Finding.","findings":[{"id":"find-finding","name":"Find","description":"Find.","questions":[{"text":"How are primitive roots found in practice, and what is known about the least primitive root?","kind":"provenance"},{"text":"Which references are standard?","kind":"provenance"}]}]}]},{"id":"use","name":"Use","description":"Applications.","layers":[{"id":"logarithm","name":"Logarithm","description":"Discrete logarithms.","findings":[{"id":"logarithm-finding","name":"Logarithm","description":"Logarithm.","questions":[{"text":"How do primitive roots define discrete logarithms and index tables?","kind":"definition"},{"text":"Which entry fits the discrete logarithm problem?","kind":"action"}]}]},{"id":"cryptography","name":"Cryptography","description":"Cryptographic use.","findings":[{"id":"cryptography-finding","name":"Cryptography","description":"Cryptography.","questions":[{"text":"How are generators used in protocols such as Diffie-Hellman, in general terms?","kind":"provenance"},{"text":"Is the user asking for parameter choices for a real system, which needs security guidance?","kind":"boundary"}]}]}]},{"id":"theory","name":"Theory","description":"Results and history.","layers":[{"id":"results","name":"Results","description":"Theorems and conjectures.","findings":[{"id":"results-finding","name":"Results","description":"Results.","questions":[{"text":"What is proved and conjectured about primitive roots, such as Artin conjecture?","kind":"provenance"},{"text":"What is proven versus conjectured?","kind":"boundary"}]}]},{"id":"history","name":"History","description":"History.","findings":[{"id":"history-finding","name":"History","description":"History.","questions":[{"text":"How did Euler, Lagrange and Gauss develop the theory of primitive roots?","kind":"provenance"},{"text":"Which entry fits Gauss?","kind":"action"}]}]}]}]},"agentConduct":{"may":["Compute and check primitive roots modulo n and explain when they exist.","Use primitive roots in teaching and in authorized cryptographic work."],"mustNot":["State that a primitive root exists for an n where it does not.","Use small or weak parameters in a real security system.","Present a conjecture, such as Artin's, as proven."],"requiresHuman":["Choosing parameters for a cryptographic system that protects people's data."]},"ethics":{"considerations":["Mistakes in number-theoretic parameters can weaken cryptography that protects people.","Mathematical credit matters to researchers."],"affectedParties":["Users of cryptographic systems","Students"]},"owners":{"steward":"Nobody owns the concept; it belongs to mathematics.","roles":[],"masterSystems":[]},"relations":[{"target":"Q4349566","type":"parent","note":"registry parent class"},{"target":"Q7366581","type":"parent","note":"registry parent class"},{"target":"root of unity modulo n","type":"related","note":"the case of maximal order"},{"target":"multiplicative group of integers modulo n","type":"related","note":"the group it generates"},{"target":"discrete logarithm","type":"related","note":"the inverse problem"},{"target":"Euler totient function","type":"related","note":"counts and orders"}],"interaction":{"identity":{"applicability":"required","items":["Vercy registry: vr.tr.primitive-root-modulo-n","Wikidata: Q948010 (https://www.wikidata.org/wiki/Q948010)","OEIS: A001918 least primitive root of the nth prime"]},"properties":{"applicability":"not-applicable","items":["number of primitive roots when they exist: phi of phi of n integers - Euler totient applied twice","smallest primitive root modulo 7: 3 integer - example"]},"recognition":{"applicability":"optional","items":["Multiplicative order equal to Euler phi of n","Exists only for n equal to 1, 2, 4, p to the k, or 2 p to the k with p an odd prime","A primitive nth root of unity in the complex numbers is a related but different object","Not visible; a property of an integer relative to a modulus."]},"capabilities":{"applicability":"required","items":["test whether an integer is a primitive root modulo n","find primitive roots for a modulus","state the existence theorem and count","explain the link to discrete logarithms"]},"hazards":{"applicability":"required","items":["Assuming a primitive root exists for every modulus","Confusing with complex roots of unity","Using small or weak generators in cryptographic settings, which belongs to security guidance"]},"interfaces":{"applicability":"required","items":["No regulation; definitions follow standard number theory texts"]},"context":{"applicability":"required","items":["Generating the multiplicative group modulo n; basis for discrete logarithms.","primitive roots modulo a prime","primitive roots modulo prime powers and twice prime powers","least primitive roots as a studied sequence","generalisations to generators of cyclic subgroups"]}},"sources":[{"title":"Wikidata item Q948010: primitive root modulo n","url":"https://www.wikidata.org/wiki/Q948010","note":"identity and sense of the item"},{"title":"Wikipedia: Primitive root modulo n","url":"https://en.wikipedia.org/wiki/Primitive_root_modulo_n","note":"general description of the item"}],"openQuestions":["Should the discrete logarithm be a separate entry?","How should OEIS sequences be linked?","How should Artin conjecture and related open problems be recorded?"],"resources":{"spec":"/models/things/publications/thing-q948010/spec.json"},"provenance":{"origin":"thing registry research (pass 2)","builtFrom":["models/things/publications/thing-q948010/spec.json"],"providers":["Claude"],"researchStatus":"unreviewed","generatedAt":"2026-09-12T20:13:51Z","builder":"tools/build_cards.py@1.0.0"},"completeness":{"sections":{"classifiers":"filled","whatItIs":"filled","purpose":"filled","distinguishingFeatures":"filled","structure":"filled","agentConduct":"filled","ethics":"filled","owners":"filled","relations":"filled","interaction.identity":"filled","interaction.properties":"not-applicable","interaction.recognition":"filled","interaction.capabilities":"filled","interaction.hazards":"filled","interaction.interfaces":"filled","interaction.context":"filled","sources":"filled"},"notes":{"_gate":"Published through the thing publication gate 1.0 on 2026-10-06T19:52:03+00:00: completeness 1.00; sense: Wikidata description; 2 live sources; review by codex: pass","interaction.properties":"Plane XCT: no invented physical properties."},"score":1.0}}