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Using this duality, we develop polynomial based techniques for understanding the bounded error ($\\textsf{BPP}$) and the weakly-unbounded error ($\\textsf{PP}$) communication complexities of $\\textsf{XOR}$ functions. We show the following.\n  A weak form of an interesting conjecture of Zhang and Shi (Quantum Information and Computation, 2009) (The full conjecture has just been reported to be independently settled by Hatami and Qian (A"},"verification_status":{"content_addressed":true,"pith_receipt":true,"author_attested":false,"weak_author_claims":0,"strong_author_claims":0,"externally_anchored":false,"storage_verified":false,"citation_signatures":0,"replication_records":0,"graph_snapshot":true,"references_resolved":false,"formal_links_present":false},"canonical_record":{"source":{"id":"1704.02537","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2017-04-08T21:27:11Z","cross_cats_sorted":[],"title_canon_sha256":"6c6d092b09c44b6c81616938e9dfc66241b0054b6b091383d577712ed4bbfd2e","abstract_canon_sha256":"010c40fcefc3164c08a1a1cd9b8e805061cbe2e6432d9b798e23403afa6e9809"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T00:46:45.187371Z","signature_b64":"tCwDHIzl/UrTu21/i+cswFGhDnx+8DDQgOzmiUKQMknsT2qBcMHbJm9lJdOQFBf5tHFS1hd3SIuDAiau35HBBA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"85f371ee1e015b9f0fb9de5f0219a8c9fdaccdd11554efa15bc5fa423af68101","last_reissued_at":"2026-05-18T00:46:45.186634Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T00:46:45.186634Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Dual polynomials and communication complexity of $\\textsf{XOR}$ functions","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Arkadev Chattopadhyay, Nikhil S. 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