{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2026:EPVGOTNJZRIWID2AHKAK33CXQP","short_pith_number":"pith:EPVGOTNJ","canonical_record":{"source":{"id":"2606.26993","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-25T13:10:04Z","cross_cats_sorted":["cs.SC"],"title_canon_sha256":"4f183bafd95c9e564b9e1ece8627bbe69ff0ed60277755862471fa567bfd045f","abstract_canon_sha256":"f76a3a82c2b054da6d94f58281c123eff97fc060c866d46a449130e5baf2c630"},"schema_version":"1.0"},"canonical_sha256":"23ea674da9cc51640f403a80adec5783e1770a88e8868eb7731f94a216608273","source":{"kind":"arxiv","id":"2606.26993","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.26993","created_at":"2026-06-26T01:16:06Z"},{"alias_kind":"arxiv_version","alias_value":"2606.26993v1","created_at":"2026-06-26T01:16:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.26993","created_at":"2026-06-26T01:16:06Z"},{"alias_kind":"pith_short_12","alias_value":"EPVGOTNJZRIW","created_at":"2026-06-26T01:16:06Z"},{"alias_kind":"pith_short_16","alias_value":"EPVGOTNJZRIWID2A","created_at":"2026-06-26T01:16:06Z"},{"alias_kind":"pith_short_8","alias_value":"EPVGOTNJ","created_at":"2026-06-26T01:16:06Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2026:EPVGOTNJZRIWID2AHKAK33CXQP","target":"record","payload":{"canonical_record":{"source":{"id":"2606.26993","kind":"arxiv","version":1},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-25T13:10:04Z","cross_cats_sorted":["cs.SC"],"title_canon_sha256":"4f183bafd95c9e564b9e1ece8627bbe69ff0ed60277755862471fa567bfd045f","abstract_canon_sha256":"f76a3a82c2b054da6d94f58281c123eff97fc060c866d46a449130e5baf2c630"},"schema_version":"1.0"},"canonical_sha256":"23ea674da9cc51640f403a80adec5783e1770a88e8868eb7731f94a216608273","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-06-26T01:16:06.160152Z","signature_b64":"G2ipHHbeUeOkohSisiRxKgbrbU5gAH8jbqzUUaI71TPMn+AzziJppYvLjfp8IZAJUDKmq90uYbEj+5GErwVJAQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"23ea674da9cc51640f403a80adec5783e1770a88e8868eb7731f94a216608273","last_reissued_at":"2026-06-26T01:16:06.159803Z","signature_status":"signed_v1","first_computed_at":"2026-06-26T01:16:06.159803Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"2606.26993","source_version":1,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-26T01:16:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HUbT3Ep08rZ7TzYZLMW4Ov69GhIvz9xZlNUQNF6WQUwsP5QMU5DYPxzqgG9wkBStuJu6Blb6bqu/0sXBy5uIAQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-26T03:10:16.506151Z"},"content_sha256":"7c227a6813f61b11d060184da993b4f650b10fcb9f5d0e0998239cc014fa9feb","schema_version":"1.0","event_id":"sha256:7c227a6813f61b11d060184da993b4f650b10fcb9f5d0e0998239cc014fa9feb"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2026:EPVGOTNJZRIWID2AHKAK33CXQP","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Deterministic and Efficient Ideal Arithmetic via Two-Element Representations","license":"http://creativecommons.org/licenses/by/4.0/","headline":"","cross_cats":["cs.SC"],"primary_cat":"math.NT","authors_text":"Qi Cheng","submitted_at":"2026-06-25T13:10:04Z","abstract_excerpt":"Given an ideal in a number field, it is desirable in many situations\n  to find two elements that generate the ideal over the ring of the\n  integers of the field. Existing algorithms are either randomized,\n  or impractical at cryptographic sizes.\n  In the paper, we present a deterministic\n  polynomial time algorithm to find the two-element representation of\n  an ideal. For a monic irreducible integral polynomial \\( f(x) \\),\n  let \\( K=\\Q[x]/(f) \\) be the number field, and\n  \\( O_K \\) be the integral closure. Our algorithm works when\n  the norm of the input ideal is co-prime to the index\n  \\( [O"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.26993","kind":"arxiv","version":1},"verdict":{"id":null,"model_set":{},"created_at":null,"strongest_claim":"","one_line_summary":"","pipeline_version":null,"weakest_assumption":"","pith_extraction_headline":""},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2606.26993/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"references":{"count":0,"sample":[],"resolved_work":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","internal_anchors":0},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"author_claims":{"count":0,"strong_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"builder_version":"pith-number-builder-2026-05-17-v1"},"verdict_id":null},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-06-26T01:16:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"eZlVdSEbx4eOaoU24QTb382pstJ7xJL2vtSX4D1TDQPSINdPa+76BxkAEYoY0kALwNH00G1Xbq5UXu8PpphBAg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-26T03:10:16.506544Z"},"content_sha256":"5ae6408d28b3c33e9e44f15b1e80647707f3a2d049b26770a14ec525bd6b28df","schema_version":"1.0","event_id":"sha256:5ae6408d28b3c33e9e44f15b1e80647707f3a2d049b26770a14ec525bd6b28df"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/EPVGOTNJZRIWID2AHKAK33CXQP/bundle.json","state_url":"https://pith.science/pith/EPVGOTNJZRIWID2AHKAK33CXQP/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/EPVGOTNJZRIWID2AHKAK33CXQP/bundle.json","status":"primary"}],"public_keys":[{"key_id":"pith-v1-2026-05","algorithm":"ed25519","format":"raw","public_key_b64":"stVStoiQhXFxp4s2pdzPNoqVNBMojDU/fJ2db5S3CbM=","public_key_hex":"b2d552b68890857171a78b36a5dccf368a953413288c353f7c9d9d6f94b709b3","fingerprint_sha256_b32_first128bits":"RVFV5Z2OI2J3ZUO7ERDEBCYNKS","fingerprint_sha256_hex":"8d4b5ee74e4693bcd1df2446408b0d54","rotates_at":null,"url":"https://pith.science/pith-signing-key.json","notes":"Pith uses this Ed25519 key to sign canonical record SHA-256 digests. Verify with: ed25519_verify(public_key, message=canonical_sha256_bytes, signature=base64decode(signature_b64))."}],"merge_version":"pith-open-graph-merge-v1","built_at":"2026-07-26T03:10:16Z","links":{"resolver":"https://pith.science/pith/EPVGOTNJZRIWID2AHKAK33CXQP","bundle":"https://pith.science/pith/EPVGOTNJZRIWID2AHKAK33CXQP/bundle.json","state":"https://pith.science/pith/EPVGOTNJZRIWID2AHKAK33CXQP/state.json","well_known_bundle":"https://pith.science/.well-known/pith/EPVGOTNJZRIWID2AHKAK33CXQP/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2026:EPVGOTNJZRIWID2AHKAK33CXQP","merge_version":"pith-open-graph-merge-v1","event_count":2,"valid_event_count":2,"invalid_event_count":0,"equivocation_count":0,"current":{"canonical_record":{"metadata":{"abstract_canon_sha256":"f76a3a82c2b054da6d94f58281c123eff97fc060c866d46a449130e5baf2c630","cross_cats_sorted":["cs.SC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-25T13:10:04Z","title_canon_sha256":"4f183bafd95c9e564b9e1ece8627bbe69ff0ed60277755862471fa567bfd045f"},"schema_version":"1.0","source":{"id":"2606.26993","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2606.26993","created_at":"2026-06-26T01:16:06Z"},{"alias_kind":"arxiv_version","alias_value":"2606.26993v1","created_at":"2026-06-26T01:16:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2606.26993","created_at":"2026-06-26T01:16:06Z"},{"alias_kind":"pith_short_12","alias_value":"EPVGOTNJZRIW","created_at":"2026-06-26T01:16:06Z"},{"alias_kind":"pith_short_16","alias_value":"EPVGOTNJZRIWID2A","created_at":"2026-06-26T01:16:06Z"},{"alias_kind":"pith_short_8","alias_value":"EPVGOTNJ","created_at":"2026-06-26T01:16:06Z"}],"graph_snapshots":[{"event_id":"sha256:5ae6408d28b3c33e9e44f15b1e80647707f3a2d049b26770a14ec525bd6b28df","target":"graph","created_at":"2026-06-26T01:16:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"graph_snapshot":{"author_claims":{"count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57","strong_count":0},"builder_version":"pith-number-builder-2026-05-17-v1","claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"formal_canon":{"evidence_count":0,"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"integrity":{"available":true,"clean":true,"detectors_run":[],"endpoint":"/pith/2606.26993/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"Given an ideal in a number field, it is desirable in many situations\n  to find two elements that generate the ideal over the ring of the\n  integers of the field. Existing algorithms are either randomized,\n  or impractical at cryptographic sizes.\n  In the paper, we present a deterministic\n  polynomial time algorithm to find the two-element representation of\n  an ideal. For a monic irreducible integral polynomial \\( f(x) \\),\n  let \\( K=\\Q[x]/(f) \\) be the number field, and\n  \\( O_K \\) be the integral closure. Our algorithm works when\n  the norm of the input ideal is co-prime to the index\n  \\( [O","authors_text":"Qi Cheng","cross_cats":["cs.SC"],"headline":"","license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-25T13:10:04Z","title":"Deterministic and Efficient Ideal Arithmetic via Two-Element Representations"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2606.26993","kind":"arxiv","version":1},"verdict":{"created_at":null,"id":null,"model_set":{},"one_line_summary":"","pipeline_version":null,"pith_extraction_headline":"","strongest_claim":"","weakest_assumption":""}},"verdict_id":null}}],"author_attestations":[],"timestamp_anchors":[],"storage_attestations":[],"citation_signatures":[],"replication_records":[],"corrections":[],"mirror_hints":[],"record_created":{"event_id":"sha256:7c227a6813f61b11d060184da993b4f650b10fcb9f5d0e0998239cc014fa9feb","target":"record","created_at":"2026-06-26T01:16:06Z","signer":{"key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signer_id":"pith.science","signer_type":"pith_registry"},"payload":{"attestation_state":"computed","canonical_record":{"metadata":{"abstract_canon_sha256":"f76a3a82c2b054da6d94f58281c123eff97fc060c866d46a449130e5baf2c630","cross_cats_sorted":["cs.SC"],"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"math.NT","submitted_at":"2026-06-25T13:10:04Z","title_canon_sha256":"4f183bafd95c9e564b9e1ece8627bbe69ff0ed60277755862471fa567bfd045f"},"schema_version":"1.0","source":{"id":"2606.26993","kind":"arxiv","version":1}},"canonical_sha256":"23ea674da9cc51640f403a80adec5783e1770a88e8868eb7731f94a216608273","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"23ea674da9cc51640f403a80adec5783e1770a88e8868eb7731f94a216608273","first_computed_at":"2026-06-26T01:16:06.159803Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-06-26T01:16:06.159803Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"G2ipHHbeUeOkohSisiRxKgbrbU5gAH8jbqzUUaI71TPMn+AzziJppYvLjfp8IZAJUDKmq90uYbEj+5GErwVJAQ==","signature_status":"signed_v1","signed_at":"2026-06-26T01:16:06.160152Z","signed_message":"canonical_sha256_bytes"},"source_id":"2606.26993","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:7c227a6813f61b11d060184da993b4f650b10fcb9f5d0e0998239cc014fa9feb","sha256:5ae6408d28b3c33e9e44f15b1e80647707f3a2d049b26770a14ec525bd6b28df"],"state_sha256":"7b10c6dd04a9bd9374c8f3fcc8f21afd3170c47c7dc05abb690ce40bca450cdf"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"xq83DRCWUgWUFZVlfoaLgwuWaKYyor0k4/LzTErCAsEFakuUMnjM4JpURmxLPVmyyvS97SD/8iFtskf2vLaCDQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-26T03:10:16.508731Z","bundle_sha256":"f4beb1acc6008438dfd125d6451b82befcf714303204d573704c6c180830292b"}}