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While its security over prime fields $\\mathbb{F}_p$ is well-documented, recent interest has shifted toward instantiations over extension fields $\\mathbb{F}_{p^r}$. This paper presents the first comprehensive cryptanalysis of the single-degree Legendre PRF operating over $\\mathbb{F}_{p^r}$.\n  First, we analyze polynomial input encoding under a standard passive"},"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":"2604.04833","kind":"arxiv","version":3},"metadata":{"license":"http://creativecommons.org/licenses/by/4.0/","primary_cat":"cs.CR","submitted_at":"2026-04-06T16:35:32Z","cross_cats_sorted":["math.NT"],"title_canon_sha256":"1611a5b4f24465b3eb769c2e3a1f19a7928f19245a8cbd2d8838ba5f8f019c2b","abstract_canon_sha256":"df8e14ee4c5322bb028108462faceb3c09a49bcde3b525e785f9bec104e05899"},"schema_version":"1.0"},"receipt":{"kind":"pith_receipt","key_id":"pith-v1-2026-05","algorithm":"ed25519","signed_at":"2026-05-25T02:01:18.932694Z","signature_b64":"9JJr+XMc/Pv+HTHhrQat80dJPyHCpeBJJJXZPAI2RGWFVFLN72ZBV+1HEAsjPRE/peeRxcn7ka6KlsOjSPTJAA==","signed_message":"canonical_sha256_bytes","builder_version":"pith-number-builder-2026-05-17-v1","receipt_version":"0.3","canonical_sha256":"72a1e0f0209e4aea2ba084e4b547c796d68f2a643322ebfbe09eff4d857cc050","last_reissued_at":"2026-05-25T02:01:18.931731Z","signature_status":"signed_v1","first_computed_at":"2026-05-25T02:01:18.931731Z","public_key_fingerprint":"8d4b5ee74e4693bcd1df2446408b0d54"},"graph_snapshot":{"paper":{"title":"Cryptanalysis of the Legendre Pseudorandom Function over Extension Fields","license":"http://creativecommons.org/licenses/by/4.0/","headline":"The single-degree Legendre PRF over extension fields admits efficient key recovery through periodic fracture bucketing and geometric-sequence collisions.","cross_cats":["math.NT"],"primary_cat":"cs.CR","authors_text":"Daksh Pandey","submitted_at":"2026-04-06T16:35:32Z","abstract_excerpt":"The Legendre Pseudorandom Function (PRF) is a highly efficient cryptographic primitive built upon the Legendre symbol, valued for its low multiplicative complexity in Multi-Party Computation (MPC) and Zero-Knowledge Proof (ZKP) protocols. While its security over prime fields $\\mathbb{F}_p$ is well-documented, recent interest has shifted toward instantiations over extension fields $\\mathbb{F}_{p^r}$. This paper presents the first comprehensive cryptanalysis of the single-degree Legendre PRF operating over $\\mathbb{F}_{p^r}$.\n  First, we analyze polynomial input encoding under a standard passive"},"claims":{"count":4,"items":[{"kind":"strongest_claim","text":"We demonstrate that an adversary can systematically group fractured sequences by their structural shapes to bypass this defense, recovering the secret key in O(U · p^r/M) operations... an adversary can circumvent the additive fracture by evaluating the PRF along a geometric sequence generated by a primitive polynomial... extract the key in O(p^r/M) operations. Finally, we establish the cryptographic boundaries of these attacks, formally proving the necessity of higher-degree key variants (d ≥ 2) to achieve exponential security against structural reduction in extension fields.","source":"verdict.strongest_claim","status":"machine_extracted","claim_id":"C1","attestation":"unclaimed"},{"kind":"weakest_assumption","text":"The analysis relies on polynomial input encoding without carry-overs producing a deterministically periodic fracture, and on the existence of primitive polynomials enabling strict multiplicative homomorphism in the active model; if real implementations use different encodings or the fracture periodicity does not hold as claimed, the bucketing and collision attacks may fail.","source":"verdict.weakest_assumption","status":"machine_extracted","claim_id":"C2","attestation":"unclaimed"},{"kind":"one_line_summary","text":"The Legendre PRF over extension fields admits key recovery in O(p^r/M) operations via differential signature bucketing on fractured sequences or geometric sequence queries exploiting multiplicative homomorphism, proving higher-degree keys are required for exponential security.","source":"verdict.one_line_summary","status":"machine_extracted","claim_id":"C3","attestation":"unclaimed"},{"kind":"headline","text":"The single-degree Legendre PRF over extension fields admits efficient key recovery through periodic fracture bucketing and geometric-sequence collisions.","source":"verdict.pith_extraction.headline","status":"machine_extracted","claim_id":"C4","attestation":"unclaimed"}],"snapshot_sha256":"806a3309f25f614e5ed00b8b36701bb8fff60c2cbb151429a9ca1f94e1bcc4b7"},"source":{"id":"2604.04833","kind":"arxiv","version":3},"verdict":{"id":"9f2ee0d4-1f78-4bbb-bac8-fa2cde1409a2","model_set":{"reader":"grok-4.3"},"created_at":"2026-05-10T18:54:10.745331Z","strongest_claim":"We demonstrate that an adversary can systematically group fractured sequences by their structural shapes to bypass this defense, recovering the secret key in O(U · p^r/M) operations... an adversary can circumvent the additive fracture by evaluating the PRF along a geometric sequence generated by a primitive polynomial... extract the key in O(p^r/M) operations. Finally, we establish the cryptographic boundaries of these attacks, formally proving the necessity of higher-degree key variants (d ≥ 2) to achieve exponential security against structural reduction in extension fields.","one_line_summary":"The Legendre PRF over extension fields admits key recovery in O(p^r/M) operations via differential signature bucketing on fractured sequences or geometric sequence queries exploiting multiplicative homomorphism, proving higher-degree keys are required for exponential security.","pipeline_version":"pith-pipeline@v0.9.0","weakest_assumption":"The analysis relies on polynomial input encoding without carry-overs producing a deterministically periodic fracture, and on the existence of primitive polynomials enabling strict multiplicative homomorphism in the active model; if real implementations use different encodings or the fracture periodicity does not hold as claimed, the bucketing and collision attacks may fail.","pith_extraction_headline":"The single-degree Legendre PRF over extension fields admits efficient key recovery through periodic fracture bucketing and geometric-sequence collisions."},"integrity":{"clean":true,"summary":{"advisory":0,"critical":0,"by_detector":{},"informational":0},"endpoint":"/pith/2604.04833/integrity.json","findings":[],"available":true,"detectors_run":[],"snapshot_sha256":"c28c3603d3b5d939e8dc4c7e95fa8dfce3d595e45f758748cecf8e644a296938"},"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":"2604.04833","created_at":"2026-05-25T02:01:18.931859+00:00"},{"alias_kind":"arxiv_version","alias_value":"2604.04833v3","created_at":"2026-05-25T02:01:18.931859+00:00"},{"alias_kind":"doi","alias_value":"10.48550/arxiv.2604.04833","created_at":"2026-05-25T02:01:18.931859+00:00"},{"alias_kind":"pith_short_12","alias_value":"OKQ6B4BATZFO","created_at":"2026-05-25T02:01:18.931859+00:00"},{"alias_kind":"pith_short_16","alias_value":"OKQ6B4BATZFOUK5A","created_at":"2026-05-25T02:01:18.931859+00:00"},{"alias_kind":"pith_short_8","alias_value":"OKQ6B4BA","created_at":"2026-05-25T02:01:18.931859+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/OKQ6B4BATZFOUK5AQTSLKR6HS3","json":"https://pith.science/pith/OKQ6B4BATZFOUK5AQTSLKR6HS3.json","graph_json":"https://pith.science/api/pith-number/OKQ6B4BATZFOUK5AQTSLKR6HS3/graph.json","events_json":"https://pith.science/api/pith-number/OKQ6B4BATZFOUK5AQTSLKR6HS3/events.json","paper":"https://pith.science/paper/OKQ6B4BA"},"agent_actions":{"view_html":"https://pith.science/pith/OKQ6B4BATZFOUK5AQTSLKR6HS3","download_json":"https://pith.science/pith/OKQ6B4BATZFOUK5AQTSLKR6HS3.json","view_paper":"https://pith.science/paper/OKQ6B4BA","resolve_alias":"https://pith.science/api/pith-number/resolve?arxiv=2604.04833&json=true","fetch_graph":"https://pith.science/api/pith-number/OKQ6B4BATZFOUK5AQTSLKR6HS3/graph.json","fetch_events":"https://pith.science/api/pith-number/OKQ6B4BATZFOUK5AQTSLKR6HS3/events.json","actions":{"anchor_timestamp":"https://pith.science/pith/OKQ6B4BATZFOUK5AQTSLKR6HS3/action/timestamp_anchor","attest_storage":"https://pith.science/pith/OKQ6B4BATZFOUK5AQTSLKR6HS3/action/storage_attestation","attest_author":"https://pith.science/pith/OKQ6B4BATZFOUK5AQTSLKR6HS3/action/author_attestation","sign_citation":"https://pith.science/pith/OKQ6B4BATZFOUK5AQTSLKR6HS3/action/citation_signature","submit_replication":"https://pith.science/pith/OKQ6B4BATZFOUK5AQTSLKR6HS3/action/replication_record"}},"created_at":"2026-05-25T02:01:18.931859+00:00","updated_at":"2026-05-25T02:01:18.931859+00:00"}