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We show that there are asymptotically the same number of Atkin and Elkies primes ell < L on average over all curves E over F_q, provided that L >= (log q)^e for any fixed e > 0 and a sufficiently large q. We use this result to design and analyse a fast algorithm to generate random elliptic curves with #E(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":"1112.3390","kind":"arxiv","version":2},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2011-12-14T23:23:02Z","cross_cats_sorted":[],"title_canon_sha256":"fd63b1eb5b82ce8550255dd48803a6bf5bd7118970496ff7184493b64409642f","abstract_canon_sha256":"2adaf55acdba7d082ad2fa6409ef94b41dde3140adbdf2d1010b88b3e75bbcb6"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T02:49:30.147396Z","signature_b64":"OCeLgg1q6Tcb1sdYfKW/fY2vNuerK0pWQqF/FmB6uVkJCwRHLGJl4WYx1/S1SM5IaleWF9CZqlHqWCETC53cAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"9cae812264e87bb5ac85e399b57f58a7bacbc2193c20d30841c2acf78cd762cd","last_reissued_at":"2026-05-18T02:49:30.146939Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T02:49:30.146939Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Distribution of Atkin and Elkies Primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.NT","authors_text":"Andrew V. Sutherland, Igor E. Shparlinski","submitted_at":"2011-12-14T23:23:02Z","abstract_excerpt":"Given an elliptic curve E over a finite field F_q of q elements, we say that an odd prime ell not dividing q is an Elkies prime for E if t_E^2 - 4q is a square modulo ell, where t_E = q+1 - #E(F_q) and #E(F_q) is the number of F_q-rational points on E; otherwise ell is called an Atkin prime. We show that there are asymptotically the same number of Atkin and Elkies primes ell < L on average over all curves E over F_q, provided that L >= (log q)^e for any fixed e > 0 and a sufficiently large q. 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