{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2015:7DT34K3MBOQKJUOBVNMCHN2SHT","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":"475a4add3f1b4c8090bd96839276733973c24f6fe40d3056edb2fbee6b370bff","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-03-27T09:30:53Z","title_canon_sha256":"12fabbfac7ca690e16f2c6526757b0962e81fe13ef7dd6c3c7e768bb8445ff3b"},"schema_version":"1.0","source":{"id":"1503.08001","kind":"arxiv","version":3}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1503.08001","created_at":"2026-05-18T01:55:52Z"},{"alias_kind":"arxiv_version","alias_value":"1503.08001v3","created_at":"2026-05-18T01:55:52Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1503.08001","created_at":"2026-05-18T01:55:52Z"},{"alias_kind":"pith_short_12","alias_value":"7DT34K3MBOQK","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_16","alias_value":"7DT34K3MBOQKJUOB","created_at":"2026-05-18T12:29:07Z"},{"alias_kind":"pith_short_8","alias_value":"7DT34K3M","created_at":"2026-05-18T12:29:07Z"}],"graph_snapshots":[{"event_id":"sha256:1718e1facba30b615bfc9e8a05587aa4e74d54aad8121509739096e25e95715f","target":"graph","created_at":"2026-05-18T01:55:52Z","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":"In these short notes, we will show the following. Let F_q be a finite field and let E/\\F_q be an elliptic curve. Let S_r be the rth summation/Semaev polynomial for E.\n  Under an assumption, we show that it is NP-complete to check if S_r evaluates to zero on some input. Unconditionally, we prove a similar result for summation polynomials over singular curves. This suggests limitations in the usage of summation polynomials in for example algorithms to solve the elliptic curve discrete logarithm problem.\n  Assume that q is a power of 2. We show that the Weil descent to F_2 of S_3 for ordinary cur","authors_text":"Michiel Kosters, Sze Ling Yeo","cross_cats":["math.AG"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-03-27T09:30:53Z","title":"Notes on summation polynomials"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1503.08001","kind":"arxiv","version":3},"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":"sha256:9a0dda39ead7030193cab2e0f740b4a1f880a59424ae40f0bdc8071d73387d29","target":"record","created_at":"2026-05-18T01:55:52Z","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":"475a4add3f1b4c8090bd96839276733973c24f6fe40d3056edb2fbee6b370bff","cross_cats_sorted":["math.AG"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2015-03-27T09:30:53Z","title_canon_sha256":"12fabbfac7ca690e16f2c6526757b0962e81fe13ef7dd6c3c7e768bb8445ff3b"},"schema_version":"1.0","source":{"id":"1503.08001","kind":"arxiv","version":3}},"canonical_sha256":"f8e7be2b6c0ba0a4d1c1ab5823b7523cf1f8f0bb9e21980fabfb50c73639f638","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"f8e7be2b6c0ba0a4d1c1ab5823b7523cf1f8f0bb9e21980fabfb50c73639f638","first_computed_at":"2026-05-18T01:55:52.514298Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T01:55:52.514298Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"D0GnvoF3RPVJV6Rh+cffy1SKEB0AfFo9CZiIK2eIW4tGk9bpTWQpV96LUWvo3cG79QscpbWeGzH9Hy/FdO/jAA==","signature_status":"signed_v1","signed_at":"2026-05-18T01:55:52.514831Z","signed_message":"canonical_sha256_bytes"},"source_id":"1503.08001","source_kind":"arxiv","source_version":3}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:9a0dda39ead7030193cab2e0f740b4a1f880a59424ae40f0bdc8071d73387d29","sha256:1718e1facba30b615bfc9e8a05587aa4e74d54aad8121509739096e25e95715f"],"state_sha256":"42a2d8748fc066745a554b5cd061d59ed6a9be3c2c4958e870385715fc31002b"}