{"bundle_type":"pith_open_graph_bundle","bundle_version":"1.0","pith_number":"pith:2016:XLG7YIMYHNP6CUV4YKTVZWU3U5","short_pith_number":"pith:XLG7YIMY","canonical_record":{"source":{"id":"1603.06173","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-03-20T02:11:32Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"d441fcef2a00c279ed5f10693fafa8082018f2b32fd9bed2c1f099fb925bd6ad","abstract_canon_sha256":"4a0366b11b3ff041c84998c8c47b4a496e169a7877d43483b1b1bef31d06b168"},"schema_version":"1.0"},"canonical_sha256":"bacdfc21983b5fe152bcc2a75cda9ba7528ae9d7e14615aba1f331714454ae30","source":{"kind":"arxiv","id":"1603.06173","version":2},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.06173","created_at":"2026-05-18T00:58:09Z"},{"alias_kind":"arxiv_version","alias_value":"1603.06173v2","created_at":"2026-05-18T00:58:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.06173","created_at":"2026-05-18T00:58:09Z"},{"alias_kind":"pith_short_12","alias_value":"XLG7YIMYHNP6","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XLG7YIMYHNP6CUV4","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XLG7YIMY","created_at":"2026-05-18T12:30:51Z"}],"events":[{"event_type":"record_created","subject_pith_number":"pith:2016:XLG7YIMYHNP6CUV4YKTVZWU3U5","target":"record","payload":{"canonical_record":{"source":{"id":"1603.06173","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-03-20T02:11:32Z","cross_cats_sorted":["math.IT"],"title_canon_sha256":"d441fcef2a00c279ed5f10693fafa8082018f2b32fd9bed2c1f099fb925bd6ad","abstract_canon_sha256":"4a0366b11b3ff041c84998c8c47b4a496e169a7877d43483b1b1bef31d06b168"},"schema_version":"1.0"},"canonical_sha256":"bacdfc21983b5fe152bcc2a75cda9ba7528ae9d7e14615aba1f331714454ae30","receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:58:09.549180Z","signature_b64":"XthtHFZsIdpu96ZTkEW9x/QUfPPpo8CEhLrwf/JkREp8Ld8W6srr0SQaWwc3PCNKAtO0ETpkv/9CFNSWPBypDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"bacdfc21983b5fe152bcc2a75cda9ba7528ae9d7e14615aba1f331714454ae30","last_reissued_at":"2026-05-18T00:58:09.548528Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:58:09.548528Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"source_kind":"arxiv","source_id":"1603.06173","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":"2026-05-18T00:58:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"v14KUjZfplpqQ8qJ14HbKRXerKXns5pwn07+SHIGlYUCyeZc9jAwO+a5Ym0mt3eQQZ6FnU6tJztzpMLpmWzTBg==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T17:20:22.299717Z"},"content_sha256":"880a2ca83717080e7f6ccb351d4a668157f319d8555d6b172abefc12d9dcf42a","schema_version":"1.0","event_id":"sha256:880a2ca83717080e7f6ccb351d4a668157f319d8555d6b172abefc12d9dcf42a"},{"event_type":"graph_snapshot","subject_pith_number":"pith:2016:XLG7YIMYHNP6CUV4YKTVZWU3U5","target":"graph","payload":{"graph_snapshot":{"paper":{"title":"The Two-Modular Fourier Transform of Binary Functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["math.IT"],"primary_cat":"cs.IT","authors_text":"Emanuele Viterbo, Jean-Claude Belfiore, Yi Hong","submitted_at":"2016-03-20T02:11:32Z","abstract_excerpt":"In this paper, we provide a solution to the open problem of computing the Fourier transform of a binary function defined over $n$-bit vectors taking $m$-bit vector values. In particular, we introduce the two-modular Fourier transform (TMFT) of a binary function $f:G\\rightarrow {\\cal R}$, where $G = (\\mathbb{F}_2^n,+)$ is the group of $n$ bit vectors with bitwise modulo two addition $+$, and ${\\cal R}$ is a finite commutative ring of characteristic $2$. Using the specific group structure of $G$ and a sequence of nested subgroups of $G$, we define the fast TMFT and its inverse. Since the image $"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06173","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":"2026-05-18T00:58:09Z","supersedes":[],"prev_event":null,"signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"yUzbnncJvpNCaxqeDCwSrCHvwPxHvYYenK+DkMsIdl78C09Jsx+ShWbee5lDPoOd5S1qN4nw6Rk81zJDt3Q3BQ==","signed_message":"open_graph_event_sha256_bytes","signed_at":"2026-05-30T17:20:22.300334Z"},"content_sha256":"35832cdd4846ea200fda507dbad65e4c5cff2b52493fabdba88e6f442c519bc5","schema_version":"1.0","event_id":"sha256:35832cdd4846ea200fda507dbad65e4c5cff2b52493fabdba88e6f442c519bc5"}],"timestamp_proofs":[],"mirror_hints":[{"mirror_type":"https","name":"Pith Resolver","base_url":"https://pith.science","bundle_url":"https://pith.science/pith/XLG7YIMYHNP6CUV4YKTVZWU3U5/bundle.json","state_url":"https://pith.science/pith/XLG7YIMYHNP6CUV4YKTVZWU3U5/state.json","well_known_bundle_url":"https://pith.science/.well-known/pith/XLG7YIMYHNP6CUV4YKTVZWU3U5/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":"2026-05-30T17:20:22Z","links":{"resolver":"https://pith.science/pith/XLG7YIMYHNP6CUV4YKTVZWU3U5","bundle":"https://pith.science/pith/XLG7YIMYHNP6CUV4YKTVZWU3U5/bundle.json","state":"https://pith.science/pith/XLG7YIMYHNP6CUV4YKTVZWU3U5/state.json","well_known_bundle":"https://pith.science/.well-known/pith/XLG7YIMYHNP6CUV4YKTVZWU3U5/bundle.json"},"state":{"state_type":"pith_open_graph_state","state_version":"1.0","pith_number":"pith:2016:XLG7YIMYHNP6CUV4YKTVZWU3U5","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":"4a0366b11b3ff041c84998c8c47b4a496e169a7877d43483b1b1bef31d06b168","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-03-20T02:11:32Z","title_canon_sha256":"d441fcef2a00c279ed5f10693fafa8082018f2b32fd9bed2c1f099fb925bd6ad"},"schema_version":"1.0","source":{"id":"1603.06173","kind":"arxiv","version":2}},"source_aliases":[{"alias_kind":"arxiv","alias_value":"1603.06173","created_at":"2026-05-18T00:58:09Z"},{"alias_kind":"arxiv_version","alias_value":"1603.06173v2","created_at":"2026-05-18T00:58:09Z"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1603.06173","created_at":"2026-05-18T00:58:09Z"},{"alias_kind":"pith_short_12","alias_value":"XLG7YIMYHNP6","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_16","alias_value":"XLG7YIMYHNP6CUV4","created_at":"2026-05-18T12:30:51Z"},{"alias_kind":"pith_short_8","alias_value":"XLG7YIMY","created_at":"2026-05-18T12:30:51Z"}],"graph_snapshots":[{"event_id":"sha256:35832cdd4846ea200fda507dbad65e4c5cff2b52493fabdba88e6f442c519bc5","target":"graph","created_at":"2026-05-18T00:58:09Z","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 this paper, we provide a solution to the open problem of computing the Fourier transform of a binary function defined over $n$-bit vectors taking $m$-bit vector values. In particular, we introduce the two-modular Fourier transform (TMFT) of a binary function $f:G\\rightarrow {\\cal R}$, where $G = (\\mathbb{F}_2^n,+)$ is the group of $n$ bit vectors with bitwise modulo two addition $+$, and ${\\cal R}$ is a finite commutative ring of characteristic $2$. Using the specific group structure of $G$ and a sequence of nested subgroups of $G$, we define the fast TMFT and its inverse. Since the image $","authors_text":"Emanuele Viterbo, Jean-Claude Belfiore, Yi Hong","cross_cats":["math.IT"],"headline":"","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-03-20T02:11:32Z","title":"The Two-Modular Fourier Transform of Binary Functions"},"references":{"count":0,"internal_anchors":0,"resolved_work":0,"sample":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1603.06173","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":"sha256:880a2ca83717080e7f6ccb351d4a668157f319d8555d6b172abefc12d9dcf42a","target":"record","created_at":"2026-05-18T00:58:09Z","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":"4a0366b11b3ff041c84998c8c47b4a496e169a7877d43483b1b1bef31d06b168","cross_cats_sorted":["math.IT"],"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.IT","submitted_at":"2016-03-20T02:11:32Z","title_canon_sha256":"d441fcef2a00c279ed5f10693fafa8082018f2b32fd9bed2c1f099fb925bd6ad"},"schema_version":"1.0","source":{"id":"1603.06173","kind":"arxiv","version":2}},"canonical_sha256":"bacdfc21983b5fe152bcc2a75cda9ba7528ae9d7e14615aba1f331714454ae30","receipt":{"algorithm":"ed25519","builder_version":"pith-number-builder-2026-05-17-v1","canonical_sha256":"bacdfc21983b5fe152bcc2a75cda9ba7528ae9d7e14615aba1f331714454ae30","first_computed_at":"2026-05-18T00:58:09.548528Z","key_id":"pith-v1-2026-05","kind":"pith_receipt","last_reissued_at":"2026-05-18T00:58:09.548528Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","receipt_version":"0.3","signature_b64":"XthtHFZsIdpu96ZTkEW9x/QUfPPpo8CEhLrwf/JkREp8Ld8W6srr0SQaWwc3PCNKAtO0ETpkv/9CFNSWPBypDg==","signature_status":"signed_v1","signed_at":"2026-05-18T00:58:09.549180Z","signed_message":"canonical_sha256_bytes"},"source_id":"1603.06173","source_kind":"arxiv","source_version":2}}},"equivocations":[],"invalid_events":[],"applied_event_ids":["sha256:880a2ca83717080e7f6ccb351d4a668157f319d8555d6b172abefc12d9dcf42a","sha256:35832cdd4846ea200fda507dbad65e4c5cff2b52493fabdba88e6f442c519bc5"],"state_sha256":"8b3c5645d69e55cad0f4f560bc1fc83cddfd2191296c03336d9f759b5b1c6503"},"bundle_signature":{"signature_status":"signed_v1","algorithm":"ed25519","key_id":"pith-v1-2026-05","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54","signature_b64":"8yc+NBUKRXltXGTcTu2ODpJ4NpM+IVZIzRlHP1QSj+lCsUQ9jANJqqWwL1bhz1VgsRvVaZWww/SXWkkUG99WDw==","signed_message":"bundle_sha256_bytes","signed_at":"2026-05-30T17:20:22.303482Z","bundle_sha256":"f3bf1d63459390f7c64d00fe5b20ac55d4671ff2e431d07a3b6a5358f95099dc"}}