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We show that the sum can be evaluated in polynomial time in n and log N when f is a quadratic polynomial. This is true even when the factorization of N is unknown. Previously, this was known for a prime modulus N. On the other hand, for very specific families of polynomials of degree \\ge 3, we show the problem is #P-hard, even f"},"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":"1005.2632","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"cs.CC","submitted_at":"2010-05-14T21:31:11Z","cross_cats_sorted":[],"title_canon_sha256":"9fe461468ebabf1016b1ef6bf140e50d7baff969992e78754e02c9fce1f0a3c9","abstract_canon_sha256":"703f6184dbd6e531832ee278430a799f3d5bc13e09b6528b548cfe23558a5051"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:07:24.560921Z","signature_b64":"1Dlnv9C3ikg/3JWX5cueujNcezq6pQoZXEIuT55OJwyRtwLbNt9vyH5sxLHzaOFThs0iIhGyolWCb1qDxVKpAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"64f04aa9b1c3847d26353e845384642870751b3bbad7e1d06613f2f55c501888","last_reissued_at":"2026-05-18T02:07:24.560467Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:07:24.560467Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On Tractable Exponential Sums","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"cs.CC","authors_text":"Jin-Yi Cai, Pinyan Lu, Richard Lipton, Xi Chen","submitted_at":"2010-05-14T21:31:11Z","abstract_excerpt":"We consider the problem of evaluating certain exponential sums. 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