{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2024:K7XSM62PLXH2VQHZX5QV4T5FWV","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":"12d9d3fd418a3ffa169b2d668722aab7fff6c9527d0fd9448acd73cb5002c9c6","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-12-06T11:37:28Z","title_canon_sha256":"6f4bdc6a14f6aa792091fdd417a526ddd5eac6f0409bba0877f9a1fc51a35e4c"},"schema_version":"1.0","source":{"id":"2412.04967","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"2412.04967","created_at":"2026-07-05T10:18:46Z"},{"alias_kind":"arxiv_version","alias_value":"2412.04967v2","created_at":"2026-07-05T10:18:46Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2412.04967","created_at":"2026-07-05T10:18:46Z"},{"alias_kind":"pith_short_12","alias_value":"K7XSM62PLXH2","created_at":"2026-07-05T10:18:46Z"},{"alias_kind":"pith_short_16","alias_value":"K7XSM62PLXH2VQHZ","created_at":"2026-07-05T10:18:46Z"},{"alias_kind":"pith_short_8","alias_value":"K7XSM62P","created_at":"2026-07-05T10:18:46Z"}],"graph_snapshots":[{"event_id":"sha256:c86c65926c4f27c936405106aa47a262d1c30e2c19f2b92a1f9022e303bf38c1","target":"graph","created_at":"2026-07-05T10:18:46Z","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/2412.04967/integrity.json","findings":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938","summary":{"advisory":0,"by_detector":{},"critical":0,"informational":0}},"paper":{"abstract_excerpt":"The Hidden Subset Sum Problem (HSSP) is a significant NP-complete problem in number theory and combinatorics, with applications in cryptography and AI privacy. For the $(n,k)$-complete HSSP, where a target multiset must be recovered from its all $k$-subset sums, existing algorithms face limitations due to high complexity or intractability. This paper proposes two deterministic algorithms: a brute-force approach, and a novel method leveraging symmetric polynomials and Vieta's formulas with $O\\left(\\sum_{u=1}^n p(u,\\leq k)^3+\\binom{n}{k}n\\right)$ complexity, where $ p(u,\\leq k)$ counts the numbe","authors_text":"Changheng Li, Lixia Luo, Qiongxiu Li","cross_cats":["math.NT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-12-06T11:37:28Z","title":"Deterministic Algorithms to Solve the $(n,k)$-Complete Hidden Subset Sum Problem"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"2412.04967","kind":"arxiv","version":2},"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":"sha256:61487222c7172ae502eaed36c0c69a99d68c52910c2f7b2e171a8d92bb6e38c3","target":"record","created_at":"2026-07-05T10:18:46Z","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":"12d9d3fd418a3ffa169b2d668722aab7fff6c9527d0fd9448acd73cb5002c9c6","cross_cats_sorted":["math.NT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.CO","submitted_at":"2024-12-06T11:37:28Z","title_canon_sha256":"6f4bdc6a14f6aa792091fdd417a526ddd5eac6f0409bba0877f9a1fc51a35e4c"},"schema_version":"1.0","source":{"id":"2412.04967","kind":"arxiv","version":2}},"canonical_sha256":"57ef267b4f5dcfaac0f9bf615e4fa5b5748e180b5f6367783951a5ebb289bc4d","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"57ef267b4f5dcfaac0f9bf615e4fa5b5748e180b5f6367783951a5ebb289bc4d","first_computed_at":"2026-07-05T10:18:46.355999Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-07-05T10:18:46.355999Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"ObEEgZz9M2DgDIFz4Io2H82ppxWnUSANCEbKp+kuVI3zXwAfMOzImomDdu/uq2hK+kWMSIzVcBSTV1/l7ipCBQ==","signature_status":"signed_v1","signed_at":"2026-07-05T10:18:46.356551Z","signed_message":"canonical_sha256_bytes"},"source_id":"2412.04967","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:61487222c7172ae502eaed36c0c69a99d68c52910c2f7b2e171a8d92bb6e38c3","sha256:c86c65926c4f27c936405106aa47a262d1c30e2c19f2b92a1f9022e303bf38c1"],"state_sha256":"4c0b2ab9bf542bc2b6d3664aadde17e548344fd898eaa3df10f3e8a58c017b1e"}