{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2013:FZL4IQCMD445ARDBRLIY5ZCRK5","short_pith_number":"pith:FZL4IQCM","canonical_record":{"source":{"id":"1304.4440","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-04-16T13:37:40Z","cross_cats_sorted":[],"title_canon_sha256":"f583bf191f578ee7daf008486b4419d0ab2b8111f87c4e4e983ea6f4cbaa2ff0","abstract_canon_sha256":"46ed3c52276736e463b0a0d1cff9e0d0cbe0386faa997956bd882c87db27d653"},"schema_version":"1.0"},"canonical_sha256":"2e57c4404c1f39d044618ad18ee451574a78c7871b00c5cd3b094c7d61601504","source":{"kind":"arxiv","id":"1304.4440","version":1},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.4440","created_at":"2026-05-18T03:27:53Z"},{"alias_kind":"arxiv_version","alias_value":"1304.4440v1","created_at":"2026-05-18T03:27:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.4440","created_at":"2026-05-18T03:27:53Z"},{"alias_kind":"pith_short_12","alias_value":"FZL4IQCMD445","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"FZL4IQCMD445ARDB","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"FZL4IQCM","created_at":"2026-05-18T12:27:45Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2013:FZL4IQCMD445ARDBRLIY5ZCRK5","target":"record","payload":{"canonical_record":{"source":{"id":"1304.4440","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-04-16T13:37:40Z","cross_cats_sorted":[],"title_canon_sha256":"f583bf191f578ee7daf008486b4419d0ab2b8111f87c4e4e983ea6f4cbaa2ff0","abstract_canon_sha256":"46ed3c52276736e463b0a0d1cff9e0d0cbe0386faa997956bd882c87db27d653"},"schema_version":"1.0"},"canonical_sha256":"2e57c4404c1f39d044618ad18ee451574a78c7871b00c5cd3b094c7d61601504","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:27:53.075715Z","signature_b64":"kRFcKvNfsADKaPGqoyJT+Ud+PRJalzf8eizmQNJZ1HOUS4TE+sRvyJIpjy0mApmtfgKdGMmr4mdIWHI86Gn5DQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"2e57c4404c1f39d044618ad18ee451574a78c7871b00c5cd3b094c7d61601504","last_reissued_at":"2026-05-18T03:27:53.075099Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:27:53.075099Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1304.4440","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":"2026-05-18T03:27:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"tVDAD8uMZQPi7suOvfxYqCbzsUR3OFAvgvgYjjrGWSS4vJdmofXDTXO30uTjIsY19VkqxGpPqFbSvRCpCyeQBQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T09:58:38.944043Z"},"content_sha256":"676c636832416f2cb812f21a3b64134b4131d9fe21a57de901b642e24baae257","schema_version":"1.0","event_id":"sha256:676c636832416f2cb812f21a3b64134b4131d9fe21a57de901b642e24baae257"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2013:FZL4IQCMD445ARDBRLIY5ZCRK5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"2- and 3-modular Lattice Wiretap Codes in Small Dimensions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Fr\\'ed\\'erique Oggier, Fuchun Lin, Patrick Sol\\'e","submitted_at":"2013-04-16T13:37:40Z","abstract_excerpt":"A recent line of work on lattice codes for Gaussian wiretap channels introduced a new lattice invariant called secrecy gain as a code design criterion which captures the confusion that lattice coding produces at an eavesdropper. Following up the study of unimodular lattice wiretap codes [1], this paper investigates 2- and 3-modular lattices and compares them with unimodular lattices. Most even 2- and 3-modular lattices are found to have better performance, that is, a higher secrecy gain than the best unimodular lattices in dimension n, n is between 2 and 23. Odd 2-modular lattices are consider"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.4440","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":"2026-05-18T03:27:53Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"HVpODjEO4ZtJoLnRQSRxh00Urjvd5edM4p6quaPx8h/uwYi114ZAMLANOPJgv4YRt/j3K4u6YRTCn/ms4WpHDQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-31T09:58:38.944695Z"},"content_sha256":"0c838d9871695520ad13215d032beb39d77d2f9951ba4a37df742b8e21c38d4b","schema_version":"1.0","event_id":"sha256:0c838d9871695520ad13215d032beb39d77d2f9951ba4a37df742b8e21c38d4b"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/FZL4IQCMD445ARDBRLIY5ZCRK5/bundle.json","state_url":"https://pith.science/pith/FZL4IQCMD445ARDBRLIY5ZCRK5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/FZL4IQCMD445ARDBRLIY5ZCRK5/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":"2026-05-31T09:58:38Z","links":{"resolver":"https://pith.science/pith/FZL4IQCMD445ARDBRLIY5ZCRK5","bundle":"https://pith.science/pith/FZL4IQCMD445ARDBRLIY5ZCRK5/bundle.json","state":"https://pith.science/pith/FZL4IQCMD445ARDBRLIY5ZCRK5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/FZL4IQCMD445ARDBRLIY5ZCRK5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2013:FZL4IQCMD445ARDBRLIY5ZCRK5","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":"46ed3c52276736e463b0a0d1cff9e0d0cbe0386faa997956bd882c87db27d653","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-04-16T13:37:40Z","title_canon_sha256":"f583bf191f578ee7daf008486b4419d0ab2b8111f87c4e4e983ea6f4cbaa2ff0"},"schema_version":"1.0","source":{"id":"1304.4440","kind":"arxiv","version":1}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1304.4440","created_at":"2026-05-18T03:27:53Z"},{"alias_kind":"arxiv_version","alias_value":"1304.4440v1","created_at":"2026-05-18T03:27:53Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1304.4440","created_at":"2026-05-18T03:27:53Z"},{"alias_kind":"pith_short_12","alias_value":"FZL4IQCMD445","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_16","alias_value":"FZL4IQCMD445ARDB","created_at":"2026-05-18T12:27:45Z"},{"alias_kind":"pith_short_8","alias_value":"FZL4IQCM","created_at":"2026-05-18T12:27:45Z"}],"graph_snapshots":[{"event_id":"sha256:0c838d9871695520ad13215d032beb39d77d2f9951ba4a37df742b8e21c38d4b","target":"graph","created_at":"2026-05-18T03:27:53Z","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":"A recent line of work on lattice codes for Gaussian wiretap channels introduced a new lattice invariant called secrecy gain as a code design criterion which captures the confusion that lattice coding produces at an eavesdropper. Following up the study of unimodular lattice wiretap codes [1], this paper investigates 2- and 3-modular lattices and compares them with unimodular lattices. Most even 2- and 3-modular lattices are found to have better performance, that is, a higher secrecy gain than the best unimodular lattices in dimension n, n is between 2 and 23. Odd 2-modular lattices are consider","authors_text":"Fr\\'ed\\'erique Oggier, Fuchun Lin, Patrick Sol\\'e","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-04-16T13:37:40Z","title":"2- and 3-modular Lattice Wiretap Codes in Small Dimensions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1304.4440","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":"sha256:676c636832416f2cb812f21a3b64134b4131d9fe21a57de901b642e24baae257","target":"record","created_at":"2026-05-18T03:27:53Z","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":"46ed3c52276736e463b0a0d1cff9e0d0cbe0386faa997956bd882c87db27d653","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-04-16T13:37:40Z","title_canon_sha256":"f583bf191f578ee7daf008486b4419d0ab2b8111f87c4e4e983ea6f4cbaa2ff0"},"schema_version":"1.0","source":{"id":"1304.4440","kind":"arxiv","version":1}},"canonical_sha256":"2e57c4404c1f39d044618ad18ee451574a78c7871b00c5cd3b094c7d61601504","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"2e57c4404c1f39d044618ad18ee451574a78c7871b00c5cd3b094c7d61601504","first_computed_at":"2026-05-18T03:27:53.075099Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T03:27:53.075099Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"kRFcKvNfsADKaPGqoyJT+Ud+PRJalzf8eizmQNJZ1HOUS4TE+sRvyJIpjy0mApmtfgKdGMmr4mdIWHI86Gn5DQ==","signature_status":"signed_v1","signed_at":"2026-05-18T03:27:53.075715Z","signed_message":"canonical_sha256_bytes"},"source_id":"1304.4440","source_kind":"arxiv","source_version":1}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:676c636832416f2cb812f21a3b64134b4131d9fe21a57de901b642e24baae257","sha256:0c838d9871695520ad13215d032beb39d77d2f9951ba4a37df742b8e21c38d4b"],"state_sha256":"656eafa181814c0ba2113df29d34195530ef0df92f244e191d80d71a204042ca"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"igVDXPX31sO/F0ec+pB+KzhtW1X4QwRpoC8tjI05RHPuS0ZWzYvt07fGrPYl2ZpvoQGaSX5v/rqeiBKslgeLAw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-31T09:58:38.948333Z","bundle_sha256":"1249f53cd36c9f0306e3a8609b04e4775df5015c1afa59bfa82057393a1ccba3"}}