{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:TWUZKSSDPJRYGDIA2GZNI7JXIM","short_pith_number":"pith:TWUZKSSD","canonical_record":{"source":{"id":"1302.4609","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CR","submitted_at":"2013-02-19T13:53:52Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"939d3628c36fb8cf71b9d50feee95d609b48ca69e62725140fec1a4c7072c61e","abstract_canon_sha256":"95b0ed36f73b5d0b168c839c5ed03ec021c143fc1aedd7be4ef29f6632eb9150"},"schema_version":"1.0"},"canonical_sha256":"9da9954a437a63830d00d1b2d47d37430a4187190317f43a8302c6376b9cc49b","source":{"kind":"arxiv","id":"1302.4609","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.4609","created_at":"2026-05-18T03:33:12Z"},{"alias_kind":"arxiv_version","alias_value":"1302.4609v1","created_at":"2026-05-18T03:33:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.4609","created_at":"2026-05-18T03:33:12Z"},{"alias_kind":"pith_short_12","alias_value":"TWUZKSSDPJRY","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_16","alias_value":"TWUZKSSDPJRYGDIA","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_8","alias_value":"TWUZKSSD","created_at":"2026-05-18T12:28:02Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:TWUZKSSDPJRYGDIA2GZNI7JXIM","target":"record","payload":{"canonical_record":{"source":{"id":"1302.4609","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CR","submitted_at":"2013-02-19T13:53:52Z","cross_cats_sorted":["cs.DS"],"title_canon_sha256":"939d3628c36fb8cf71b9d50feee95d609b48ca69e62725140fec1a4c7072c61e","abstract_canon_sha256":"95b0ed36f73b5d0b168c839c5ed03ec021c143fc1aedd7be4ef29f6632eb9150"},"schema_version":"1.0"},"canonical_sha256":"9da9954a437a63830d00d1b2d47d37430a4187190317f43a8302c6376b9cc49b","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:33:12.748157Z","signature_b64":"WNMhw84UKNgfMpm4kJCJWFxxSeWjdDCIJKfYlml7DrZSH3NIRTUNdC1ThycDxNnEaddEM9NPtXjWmo/+o+26Ag==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9da9954a437a63830d00d1b2d47d37430a4187190317f43a8302c6376b9cc49b","last_reissued_at":"2026-05-18T03:33:12.747354Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:33:12.747354Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1302.4609","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":"2026-05-18T03:33:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"MPt08875ubLrtkm09nq6EIGcvog6CeESihKdR5OJ/DkXKLPFr5rn81jelMM3wF654+sIKRWXaaNcYntUGHLADg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T18:55:55.342704Z"},"content_sha256":"e6b8ae18027ab141ec7752e22307d9dbb943aa0674eab118881379905ff6eaf8","schema_version":"1.0","event_id":"sha256:e6b8ae18027ab141ec7752e22307d9dbb943aa0674eab118881379905ff6eaf8"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:TWUZKSSDPJRYGDIA2GZNI7JXIM","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"Optimal information rate of secret sharing schemes on trees","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.DS"],"primary_cat":"cs.CR","authors_text":"G. Tardos, L. Csirmaz","submitted_at":"2013-02-19T13:53:52Z","abstract_excerpt":"The information rate for an access structure is the reciprocal of the load of the optimal secret sharing scheme for this structure. We determine this value for all trees: it is 1/(2-1/c), where c is the size of the largest core of the tree. A subset of the vertices of a tree is a core if it induces a connected subgraph and for each vertex in the subset one finds a neighbor outside the subset. Our result follows from a lower and an upper bound on the information rate that applies for any graph and happen to coincide for trees because of a correspondence between the size of the largest core and "},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.4609","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":"2026-05-18T03:33:12Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"o3wDFxG+wWOgmalGRgcT/3JgtVV4wmdHnVdB2MBQSSikhCqbdZHUH23PMzgZ1Jb8jKceIBgoOoKuNFPc05kUCw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-07-01T18:55:55.343047Z"},"content_sha256":"275c5938753eeb493036e0560c1c4263cb25a6980a064df1acc73a6304486dcd","schema_version":"1.0","event_id":"sha256:275c5938753eeb493036e0560c1c4263cb25a6980a064df1acc73a6304486dcd"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/TWUZKSSDPJRYGDIA2GZNI7JXIM/bundle.json","state_url":"https://pith.science/pith/TWUZKSSDPJRYGDIA2GZNI7JXIM/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/TWUZKSSDPJRYGDIA2GZNI7JXIM/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":"2026-07-01T18:55:55Z","links":{"resolver":"https://pith.science/pith/TWUZKSSDPJRYGDIA2GZNI7JXIM","bundle":"https://pith.science/pith/TWUZKSSDPJRYGDIA2GZNI7JXIM/bundle.json","state":"https://pith.science/pith/TWUZKSSDPJRYGDIA2GZNI7JXIM/state.json","well_known_bundle":"https://pith.science/.well-known/pith/TWUZKSSDPJRYGDIA2GZNI7JXIM/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:TWUZKSSDPJRYGDIA2GZNI7JXIM","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":"95b0ed36f73b5d0b168c839c5ed03ec021c143fc1aedd7be4ef29f6632eb9150","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CR","submitted_at":"2013-02-19T13:53:52Z","title_canon_sha256":"939d3628c36fb8cf71b9d50feee95d609b48ca69e62725140fec1a4c7072c61e"},"schema_version":"1.0","source":{"id":"1302.4609","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1302.4609","created_at":"2026-05-18T03:33:12Z"},{"alias_kind":"arxiv_version","alias_value":"1302.4609v1","created_at":"2026-05-18T03:33:12Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1302.4609","created_at":"2026-05-18T03:33:12Z"},{"alias_kind":"pith_short_12","alias_value":"TWUZKSSDPJRY","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_16","alias_value":"TWUZKSSDPJRYGDIA","created_at":"2026-05-18T12:28:02Z"},{"alias_kind":"pith_short_8","alias_value":"TWUZKSSD","created_at":"2026-05-18T12:28:02Z"}],"graph_snapshots":[{"event_id":"sha256:275c5938753eeb493036e0560c1c4263cb25a6980a064df1acc73a6304486dcd","target":"graph","created_at":"2026-05-18T03:33:12Z","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 information rate for an access structure is the reciprocal of the load of the optimal secret sharing scheme for this structure. We determine this value for all trees: it is 1/(2-1/c), where c is the size of the largest core of the tree. A subset of the vertices of a tree is a core if it induces a connected subgraph and for each vertex in the subset one finds a neighbor outside the subset. Our result follows from a lower and an upper bound on the information rate that applies for any graph and happen to coincide for trees because of a correspondence between the size of the largest core and ","authors_text":"G. Tardos, L. Csirmaz","cross_cats":["cs.DS"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CR","submitted_at":"2013-02-19T13:53:52Z","title":"Optimal information rate of secret sharing schemes on trees"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1302.4609","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:e6b8ae18027ab141ec7752e22307d9dbb943aa0674eab118881379905ff6eaf8","target":"record","created_at":"2026-05-18T03:33:12Z","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":"95b0ed36f73b5d0b168c839c5ed03ec021c143fc1aedd7be4ef29f6632eb9150","cross_cats_sorted":["cs.DS"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CR","submitted_at":"2013-02-19T13:53:52Z","title_canon_sha256":"939d3628c36fb8cf71b9d50feee95d609b48ca69e62725140fec1a4c7072c61e"},"schema_version":"1.0","source":{"id":"1302.4609","kind":"arxiv","version":1}},"canonical_sha256":"9da9954a437a63830d00d1b2d47d37430a4187190317f43a8302c6376b9cc49b","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"9da9954a437a63830d00d1b2d47d37430a4187190317f43a8302c6376b9cc49b","first_computed_at":"2026-05-18T03:33:12.747354Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:33:12.747354Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"WNMhw84UKNgfMpm4kJCJWFxxSeWjdDCIJKfYlml7DrZSH3NIRTUNdC1ThycDxNnEaddEM9NPtXjWmo/+o+26Ag==","signature_status":"signed_v1","signed_at":"2026-05-18T03:33:12.748157Z","signed_message":"canonical_sha256_bytes"},"source_id":"1302.4609","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:e6b8ae18027ab141ec7752e22307d9dbb943aa0674eab118881379905ff6eaf8","sha256:275c5938753eeb493036e0560c1c4263cb25a6980a064df1acc73a6304486dcd"],"state_sha256":"3b1b1e18a3d271caa4692b29cfca9402f47cd52feb21c57d83688bbac4c63f58"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"x4uNtCv3szDfQnktg+CHssk5b41d5uaC3S+9+SD33hApCEG+32wxPSPfkyL5vNBei3g81B6NvTeF/TS1l8+LCQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-07-01T18:55:55.345449Z","bundle_sha256":"6b2d58a140d0a55cfc9371996de30946ee00e0b3aeb378692a07ace98ac97a9b"}}