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These primes are used in the presently most efficient algorithm to compute $#E(\\F_q)$. In particular, the bound $L_q(E)$ such that the product of all Elkies primes for $E$ up to $L_q(E)$ exceeds $4q^{1/2}$ is a crucial parameter of this algorithm. We show that there are infinitely many pairs $(p, E)$ of primes $p$"},"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":"1301.0035","kind":"arxiv","version":1},"metadata":{"license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","primary_cat":"math.NT","submitted_at":"2013-01-01T01:35:10Z","cross_cats_sorted":["cs.CR"],"title_canon_sha256":"fc0a18f71b804e84246cc0bbbfc34ca2b8afee1a26b16004062725bc6d0d5b51","abstract_canon_sha256":"dbca0cf622df915a9ebc8dcf7d801b28f651b3d8a45e2804429e8d94b6a52c27"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-18T03:37:28.518363Z","signature_b64":"6TCFGy19e3SHQr6uh0BWLbCSB0U3iTmBFHhFeeP8EwGAtAJ4hH0/Tgb253Xw/fSmjbGSqpHnI2EF9Ox1wGjhDg==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"8db201f3cf77878e79921a036bc35555d14f4ed18c2e7adac7675444c961e45d","last_reissued_at":"2026-05-18T03:37:28.517309Z","signature_status":"signed_v1","first_computed_at":"2026-05-18T03:37:28.517309Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"On the Product of Small Elkies Primes","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":["cs.CR"],"primary_cat":"math.NT","authors_text":"Igor Shparlinski","submitted_at":"2013-01-01T01:35:10Z","abstract_excerpt":"Given an elliptic curve $E$ over a finite field $\\F_q$ of $q$ elements, we say that an odd prime $\\ell \\nmid q$ is an Elkies prime for $E$ if $t_E^2 - 4q$ is a quadratic residue modulo $\\ell$, where $t_E = q+1 - #E(\\F_q)$ and $#E(\\F_q)$ is the number of $\\F_q$-rational points on $E$. These primes are used in the presently most efficient algorithm to compute $#E(\\F_q)$. In particular, the bound $L_q(E)$ such that the product of all Elkies primes for $E$ up to $L_q(E)$ exceeds $4q^{1/2}$ is a crucial parameter of this algorithm. We show that there are infinitely many pairs $(p, E)$ of primes $p$"},"claims":{"count":0,"items":[],"snapshot_sha256":"258153158e38e3291e3d48162225fcdb2d5a3ed65a07baac614ab91432fd4f57"},"source":{"id":"1301.0035","kind":"arxiv","version":1},"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"},"aliases":[{"alias_kind":"arxiv","alias_value":"1301.0035","created_at":"2026-05-18T03:37:28.517676+00:00"},{"alias_kind":"arxiv_version","alias_value":"1301.0035v1","created_at":"2026-05-18T03:37:28.517676+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.1301.0035","created_at":"2026-05-18T03:37:28.517676+00:00"},{"alias_kind":"pith_short_12","alias_value":"RWZAD46PO6DY","created_at":"2026-05-18T12:27:59.945178+00:00"},{"alias_kind":"pith_short_16","alias_value":"RWZAD46PO6DY46MS","created_at":"2026-05-18T12:27:59.945178+00:00"},{"alias_kind":"pith_short_8","alias_value":"RWZAD46P","created_at":"2026-05-18T12:27:59.945178+00:00"}],"events":[],"event_summary":{},"paper_claims":[],"inbound_citations":{"count":0,"internal_anchor_count":0,"sample":[]},"formal_canon":{"evidence_count":0,"sample":[],"anchors":[]},"links":{"html":"https://pith.science/pith/RWZAD46PO6DY46MSDIBWXQ2VKX","json":"https://pith.science/pith/RWZAD46PO6DY46MSDIBWXQ2VKX.json","graph_json":"https://pith.science/api/pith-number/RWZAD46PO6DY46MSDIBWXQ2VKX/graph.json","events_json":"https://pith.science/api/pith-number/RWZAD46PO6DY46MSDIBWXQ2VKX/events.json","paper":"https://pith.science/paper/RWZAD46P"},"agent_actions":{"view_html":"https://pith.science/pith/RWZAD46PO6DY46MSDIBWXQ2VKX","download_json":"https://pith.science/pith/RWZAD46PO6DY46MSDIBWXQ2VKX.json","view_paper":"https://pith.science/paper/RWZAD46P","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=1301.0035&json=true","fetch_graph":"https://pith.science/api/pith-number/RWZAD46PO6DY46MSDIBWXQ2VKX/graph.json","fetch_events":"https://pith.science/api/pith-number/RWZAD46PO6DY46MSDIBWXQ2VKX/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/RWZAD46PO6DY46MSDIBWXQ2VKX/action/timestamp_anchor","attest_storage":"https://pith.science/pith/RWZAD46PO6DY46MSDIBWXQ2VKX/action/storage_attestation","attest_author":"https://pith.science/pith/RWZAD46PO6DY46MSDIBWXQ2VKX/action/author_attestation","sign_citation":"https://pith.science/pith/RWZAD46PO6DY46MSDIBWXQ2VKX/action/citation_signature","submit_replication":"https://pith.science/pith/RWZAD46PO6DY46MSDIBWXQ2VKX/action/replication_record"}},"created_at":"2026-05-18T03:37:28.517676+00:00","updated_at":"2026-05-18T03:37:28.517676+00:00"}