{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2011:MK6ZFOECJRLDITZ4K5KGI2H35Z","short_pith_number":"pith:MK6ZFOEC","canonical_record":{"source":{"id":"1111.2121","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CR","submitted_at":"2011-11-09T07:47:28Z","cross_cats_sorted":[],"title_canon_sha256":"039b46fda1b4475160dc51818d65cfb7c556c672a3f694609957fa10acc9f063","abstract_canon_sha256":"62198c2ff89d42672a6846f7aac59df541a2e85664da100d846ba3faf5cc2e7e"},"schema_version":"1.0"},"canonical_sha256":"62bd92b8824c56344f3c57546468fbee514891436d3c7b2dfb804a85ec070946","source":{"kind":"arxiv","id":"1111.2121","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.2121","created_at":"2026-05-18T04:03:06Z"},{"alias_kind":"arxiv_version","alias_value":"1111.2121v2","created_at":"2026-05-18T04:03:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.2121","created_at":"2026-05-18T04:03:06Z"},{"alias_kind":"pith_short_12","alias_value":"MK6ZFOECJRLD","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"MK6ZFOECJRLDITZ4","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"MK6ZFOEC","created_at":"2026-05-18T12:26:34Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2011:MK6ZFOECJRLDITZ4K5KGI2H35Z","target":"record","payload":{"canonical_record":{"source":{"id":"1111.2121","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CR","submitted_at":"2011-11-09T07:47:28Z","cross_cats_sorted":[],"title_canon_sha256":"039b46fda1b4475160dc51818d65cfb7c556c672a3f694609957fa10acc9f063","abstract_canon_sha256":"62198c2ff89d42672a6846f7aac59df541a2e85664da100d846ba3faf5cc2e7e"},"schema_version":"1.0"},"canonical_sha256":"62bd92b8824c56344f3c57546468fbee514891436d3c7b2dfb804a85ec070946","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T04:03:06.465573Z","signature_b64":"uL+GRKnh1ZSb4dr/hF/bDRx5tQIf/HSAYkXEDWrN6ReOehYNRGOZxc5hUCUYPcDakI6tlUM0KG8/aWS9rAtZCQ==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"62bd92b8824c56344f3c57546468fbee514891436d3c7b2dfb804a85ec070946","last_reissued_at":"2026-05-18T04:03:06.464810Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T04:03:06.464810Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1111.2121","source_version":2,"attestation_state":"computed"},"signer":{"signer_id":"pith.science","signer_type":"pith_registry","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"created_at":"2026-05-18T04:03:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"ViR7LDKj7a1NpokSQ4E7qrwr8ER6jV9h8vk3gBI2Da5axBj/w8kcxi5U5qvdlr1PpFI9Vf3RP6WnXYPK5ITzAw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T16:52:55.727674Z"},"content_sha256":"195eb0b8ba51b50f0d975e746e2e9757d703bca2e0c2f7e12bd6360850594797","schema_version":"1.0","event_id":"sha256:195eb0b8ba51b50f0d975e746e2e9757d703bca2e0c2f7e12bd6360850594797"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2011:MK6ZFOECJRLDITZ4K5KGI2H35Z","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"On $2k$-Variable Symmetric Boolean Functions with Maximum Algebraic Immunity $k$","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CR","authors_text":"Haibin Kan, Hui Wang, Jie Peng, Yuan Li","submitted_at":"2011-11-09T07:47:28Z","abstract_excerpt":"Algebraic immunity of Boolean function $f$ is defined as the minimal degree of a nonzero $g$ such that $fg=0$ or $(f+1)g=0$. Given a positive even integer $n$, it is found that the weight distribution of any $n$-variable symmetric Boolean function with maximum algebraic immunity $\\frac{n}{2}$ is determined by the binary expansion of $n$. Based on the foregoing, all $n$-variable symmetric Boolean functions with maximum algebraic immunity are constructed. The amount is $(2\\wt(n)+1)2^{\\lfloor \\log_2 n \\rfloor}$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2121","kind":"arxiv","version":2},"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":"2026-05-18T04:03:06Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"D41YSlh3np1xZpFe8eLKHYULWvZz59F+KOsqugnlBus89wVo7MdsUcVDPIFMOiP8RwCw5+0lWXTZMyt9Ql6UDw==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-26T16:52:55.728020Z"},"content_sha256":"63c07453a52abb9fdd2dc4855328da23de0a67e83886b03d1e9baa20d0447ff2","schema_version":"1.0","event_id":"sha256:63c07453a52abb9fdd2dc4855328da23de0a67e83886b03d1e9baa20d0447ff2"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/MK6ZFOECJRLDITZ4K5KGI2H35Z/bundle.json","state_url":"https://pith.science/pith/MK6ZFOECJRLDITZ4K5KGI2H35Z/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/MK6ZFOECJRLDITZ4K5KGI2H35Z/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":"2026-05-26T16:52:55Z","links":{"resolver":"https://pith.science/pith/MK6ZFOECJRLDITZ4K5KGI2H35Z","bundle":"https://pith.science/pith/MK6ZFOECJRLDITZ4K5KGI2H35Z/bundle.json","state":"https://pith.science/pith/MK6ZFOECJRLDITZ4K5KGI2H35Z/state.json","well_known_bundle":"https://pith.science/.well-known/pith/MK6ZFOECJRLDITZ4K5KGI2H35Z/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2011:MK6ZFOECJRLDITZ4K5KGI2H35Z","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":"62198c2ff89d42672a6846f7aac59df541a2e85664da100d846ba3faf5cc2e7e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CR","submitted_at":"2011-11-09T07:47:28Z","title_canon_sha256":"039b46fda1b4475160dc51818d65cfb7c556c672a3f694609957fa10acc9f063"},"schema_version":"1.0","source":{"id":"1111.2121","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1111.2121","created_at":"2026-05-18T04:03:06Z"},{"alias_kind":"arxiv_version","alias_value":"1111.2121v2","created_at":"2026-05-18T04:03:06Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1111.2121","created_at":"2026-05-18T04:03:06Z"},{"alias_kind":"pith_short_12","alias_value":"MK6ZFOECJRLD","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_16","alias_value":"MK6ZFOECJRLDITZ4","created_at":"2026-05-18T12:26:34Z"},{"alias_kind":"pith_short_8","alias_value":"MK6ZFOEC","created_at":"2026-05-18T12:26:34Z"}],"graph_snapshots":[{"event_id":"sha256:63c07453a52abb9fdd2dc4855328da23de0a67e83886b03d1e9baa20d0447ff2","target":"graph","created_at":"2026-05-18T04:03:06Z","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":"Algebraic immunity of Boolean function $f$ is defined as the minimal degree of a nonzero $g$ such that $fg=0$ or $(f+1)g=0$. Given a positive even integer $n$, it is found that the weight distribution of any $n$-variable symmetric Boolean function with maximum algebraic immunity $\\frac{n}{2}$ is determined by the binary expansion of $n$. Based on the foregoing, all $n$-variable symmetric Boolean functions with maximum algebraic immunity are constructed. The amount is $(2\\wt(n)+1)2^{\\lfloor \\log_2 n \\rfloor}$","authors_text":"Haibin Kan, Hui Wang, Jie Peng, Yuan Li","cross_cats":[],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CR","submitted_at":"2011-11-09T07:47:28Z","title":"On $2k$-Variable Symmetric Boolean Functions with Maximum Algebraic Immunity $k$"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1111.2121","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":"sha256:195eb0b8ba51b50f0d975e746e2e9757d703bca2e0c2f7e12bd6360850594797","target":"record","created_at":"2026-05-18T04:03:06Z","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":"62198c2ff89d42672a6846f7aac59df541a2e85664da100d846ba3faf5cc2e7e","cross_cats_sorted":[],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CR","submitted_at":"2011-11-09T07:47:28Z","title_canon_sha256":"039b46fda1b4475160dc51818d65cfb7c556c672a3f694609957fa10acc9f063"},"schema_version":"1.0","source":{"id":"1111.2121","kind":"arxiv","version":2}},"canonical_sha256":"62bd92b8824c56344f3c57546468fbee514891436d3c7b2dfb804a85ec070946","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"62bd92b8824c56344f3c57546468fbee514891436d3c7b2dfb804a85ec070946","first_computed_at":"2026-05-18T04:03:06.464810Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T04:03:06.464810Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"uL+GRKnh1ZSb4dr/hF/bDRx5tQIf/HSAYkXEDWrN6ReOehYNRGOZxc5hUCUYPcDakI6tlUM0KG8/aWS9rAtZCQ==","signature_status":"signed_v1","signed_at":"2026-05-18T04:03:06.465573Z","signed_message":"canonical_sha256_bytes"},"source_id":"1111.2121","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:195eb0b8ba51b50f0d975e746e2e9757d703bca2e0c2f7e12bd6360850594797","sha256:63c07453a52abb9fdd2dc4855328da23de0a67e83886b03d1e9baa20d0447ff2"],"state_sha256":"bc70c78c05a8fa5c980916c0abda0eb07039ffdc88f5b031ad6630e495ab74d0"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"az6KjhLOKsUReFtJAFP8dnt3WnQz76W8mpRS1qUiHFiBO2CUyq+l6eoGlHd/2mmlr7NF3v4vE2/xfupC6K41AQ==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-26T16:52:55.730111Z","bundle_sha256":"bbb37bd52814fa1f57ac0078151fa67e9d96b33065aedd4798aef9a51775a65c"}}