{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:32ZIJ3WW72VSERM2JDGRLLF7K3","short_pith_number":"pith:32ZIJ3WW","canonical_record":{"source":{"id":"1601.03665","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-14T16:59:43Z","cross_cats_sorted":[],"title_canon_sha256":"589fcd8fc1b4fd8ffdd656fa2994a54634a563654b88ea95f008a713492bc805","abstract_canon_sha256":"6a03be829eabb64e09bbc6b0a2894d47435cc0d6d539803a1f57f0f611854d49"},"schema_version":"1.0"},"canonical_sha256":"deb284eed6feab22459a48cd15acbf56f7b58935df5d9ffaebb9bb0c5bc455e5","source":{"kind":"arxiv","id":"1601.03665","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.03665","created_at":"2026-05-18T01:22:52Z"},{"alias_kind":"arxiv_version","alias_value":"1601.03665v1","created_at":"2026-05-18T01:22:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.03665","created_at":"2026-05-18T01:22:52Z"},{"alias_kind":"pith_short_12","alias_value":"32ZIJ3WW72VS","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"32ZIJ3WW72VSERM2","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"32ZIJ3WW","created_at":"2026-05-18T12:29:55Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:32ZIJ3WW72VSERM2JDGRLLF7K3","target":"record","payload":{"canonical_record":{"source":{"id":"1601.03665","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-14T16:59:43Z","cross_cats_sorted":[],"title_canon_sha256":"589fcd8fc1b4fd8ffdd656fa2994a54634a563654b88ea95f008a713492bc805","abstract_canon_sha256":"6a03be829eabb64e09bbc6b0a2894d47435cc0d6d539803a1f57f0f611854d49"},"schema_version":"1.0"},"canonical_sha256":"deb284eed6feab22459a48cd15acbf56f7b58935df5d9ffaebb9bb0c5bc455e5","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T01:22:52.897034Z","signature_b64":"0G+S49ryohvMAHhkgQRmHLfeJGbQjcW1kNytNHWMYXyfnY0YZ6jzoFRDQNiuZE/ExP6jN+Wpjp1QqRMvofmBDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"deb284eed6feab22459a48cd15acbf56f7b58935df5d9ffaebb9bb0c5bc455e5","last_reissued_at":"2026-05-18T01:22:52.896449Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T01:22:52.896449Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1601.03665","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":"2026-05-18T01:22:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"kPvZAh3VOd628HGJjNQVAhrTrmgavJ1Xv+8wqk/okgA1dxrL8Rt4yniAI9QNdueSJWFiuT15UDEm8hUXhyZyCA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T20:27:56.980050Z"},"content_sha256":"6fd2c5c8d19120c9f0b67475d1eb6a1afd71923a939bd386c3486b964a4e0052","schema_version":"1.0","event_id":"sha256:6fd2c5c8d19120c9f0b67475d1eb6a1afd71923a939bd386c3486b964a4e0052"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:32ZIJ3WW72VSERM2JDGRLLF7K3","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The geometry of efficient arithmetic on elliptic curves","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"David Kohel","submitted_at":"2016-01-14T16:59:43Z","abstract_excerpt":"The arithmetic of elliptic curves, namely polynomial addition and scalar multiplication, can be described in terms of global sections of line bundles on $E\\times E$ and $E$, respectively, with respect to a given projective embedding of $E$ in $\\mathbb{P}^r$. By means of a study of the finite dimensional vector spaces of global sections, we reduce the problem of constructing and finding efficiently computable polynomial maps defining the addition morphism or isogenies to linear algebra. We demonstrate the effectiveness of the method by improving the best known complexity for doubling and tripli"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03665","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":""},"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":"2026-05-18T01:22:52Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"/UQMkLaIPvVmDKRf452f6cxAHd7QOg8Wh2yr8eyEM9kiO4zIDNaz9b/sgecytHlDWVFj5+MUz3avFoC81ZolBA==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-06-02T20:27:56.980405Z"},"content_sha256":"a8a312b2fb39523473a0491f2a59fa5178d95b5d3f96fe6a20ea67dc3c3a6543","schema_version":"1.0","event_id":"sha256:a8a312b2fb39523473a0491f2a59fa5178d95b5d3f96fe6a20ea67dc3c3a6543"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/32ZIJ3WW72VSERM2JDGRLLF7K3/bundle.json","state_url":"https://pith.science/pith/32ZIJ3WW72VSERM2JDGRLLF7K3/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/32ZIJ3WW72VSERM2JDGRLLF7K3/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":"2026-06-02T20:27:56Z","links":{"resolver":"https://pith.science/pith/32ZIJ3WW72VSERM2JDGRLLF7K3","bundle":"https://pith.science/pith/32ZIJ3WW72VSERM2JDGRLLF7K3/bundle.json","state":"https://pith.science/pith/32ZIJ3WW72VSERM2JDGRLLF7K3/state.json","well_known_bundle":"https://pith.science/.well-known/pith/32ZIJ3WW72VSERM2JDGRLLF7K3/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:32ZIJ3WW72VSERM2JDGRLLF7K3","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":"6a03be829eabb64e09bbc6b0a2894d47435cc0d6d539803a1f57f0f611854d49","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-14T16:59:43Z","title_canon_sha256":"589fcd8fc1b4fd8ffdd656fa2994a54634a563654b88ea95f008a713492bc805"},"schema_version":"1.0","source":{"id":"1601.03665","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1601.03665","created_at":"2026-05-18T01:22:52Z"},{"alias_kind":"arxiv_version","alias_value":"1601.03665v1","created_at":"2026-05-18T01:22:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1601.03665","created_at":"2026-05-18T01:22:52Z"},{"alias_kind":"pith_short_12","alias_value":"32ZIJ3WW72VS","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_16","alias_value":"32ZIJ3WW72VSERM2","created_at":"2026-05-18T12:29:55Z"},{"alias_kind":"pith_short_8","alias_value":"32ZIJ3WW","created_at":"2026-05-18T12:29:55Z"}],"graph_snapshots":[{"event_id":"sha256:a8a312b2fb39523473a0491f2a59fa5178d95b5d3f96fe6a20ea67dc3c3a6543","target":"graph","created_at":"2026-05-18T01:22:52Z","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"},"paper":{"abstract_excerpt":"The arithmetic of elliptic curves, namely polynomial addition and scalar multiplication, can be described in terms of global sections of line bundles on $E\\times E$ and $E$, respectively, with respect to a given projective embedding of $E$ in $\\mathbb{P}^r$. By means of a study of the finite dimensional vector spaces of global sections, we reduce the problem of constructing and finding efficiently computable polynomial maps defining the addition morphism or isogenies to linear algebra. We demonstrate the effectiveness of the method by improving the best known complexity for doubling and tripli","authors_text":"David Kohel","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-14T16:59:43Z","title":"The geometry of efficient arithmetic on elliptic curves"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1601.03665","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":"sha256:6fd2c5c8d19120c9f0b67475d1eb6a1afd71923a939bd386c3486b964a4e0052","target":"record","created_at":"2026-05-18T01:22:52Z","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":"6a03be829eabb64e09bbc6b0a2894d47435cc0d6d539803a1f57f0f611854d49","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2016-01-14T16:59:43Z","title_canon_sha256":"589fcd8fc1b4fd8ffdd656fa2994a54634a563654b88ea95f008a713492bc805"},"schema_version":"1.0","source":{"id":"1601.03665","kind":"arxiv","version":1}},"canonical_sha256":"deb284eed6feab22459a48cd15acbf56f7b58935df5d9ffaebb9bb0c5bc455e5","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"deb284eed6feab22459a48cd15acbf56f7b58935df5d9ffaebb9bb0c5bc455e5","first_computed_at":"2026-05-18T01:22:52.896449Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:22:52.896449Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"0G+S49ryohvMAHhkgQRmHLfeJGbQjcW1kNytNHWMYXyfnY0YZ6jzoFRDQNiuZE/ExP6jN+Wpjp1QqRMvofmBDg==","signature_status":"signed_v1","signed_at":"2026-05-18T01:22:52.897034Z","signed_message":"canonical_sha256_bytes"},"source_id":"1601.03665","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:6fd2c5c8d19120c9f0b67475d1eb6a1afd71923a939bd386c3486b964a4e0052","sha256:a8a312b2fb39523473a0491f2a59fa5178d95b5d3f96fe6a20ea67dc3c3a6543"],"state_sha256":"532ad324dd2911d0863390afc3ead1afbb154796092412f468ce7e87cecc5407"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"CsfOxqVwrKrL4xDvsTFJtZ9kL5oqBGsqyM+PzNN7nvxp5yXHCYt37Zswcp3/toOpSg//Jlr167bE5isnW2gzCA==","signed_message":"bundle_sha256_bytes","signed_at":"2026-06-02T20:27:56.982356Z","bundle_sha256":"44cbb3377ed8ed230ff6da3301a3e0bb01bede42a357f5c39a16118ba786505d"}}