{"paper":{"title":"Pinpointing Super-Quadratic Quantum Enumeration Speedups: Exact and Certified Evaluation of the Guessing-Moment Exponent under Product-Distribution Advice","arxiv_id":"2609.28226","primary_cat":"cs.CR","submitted_at":"2026-09-23T14:53:56+00:00","version_compiled":"v1"},"notice":"Every quoted sentence is the authors' text from the arXiv listing abstract. Labels and probabilities are the outputs of classifier-1.13.0 on the question bank abstract-p1-20260923, exactly as returned. No sentence in this record was written by a language model.","schema":"pith.compiled-record/v0","status":"compiled","abstract":{"source":"arXiv listing abstract v1","sentences":[{"i":0,"p":1.0,"id":"s0","own":false,"role":"background","text":"Grover's algorithm gives an optimal quadratic query advantage for black-box search.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.03,"main_claim_p":0.02,"quantitative":true,"quantitative_p":0.71,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":1,"p":1.0,"id":"s1","own":false,"role":"background","text":"In cryptanalysis, however, the search often comes with additional probabilistic advice over the candidates, frequently of product form, e.g. from side-channel leakage on independent key coordinates.","hedge":{"p":0.77,"label":"asserted"},"own_p":0.05,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":2,"p":1.0,"id":"s2","own":false,"role":"background","text":"Classically, guessing in likelihood order is optimal in expectation.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.04,"main_claim_p":0.02,"quantitative":false,"quantitative_p":0.18,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":3,"p":1.0,"id":"s3","own":false,"role":"background","text":"In the quantum setting, Montanaro showed how to achieve an optimal expected query complexity, beating plain Grover on every non-uniform advice distribution (up to a constant overhead factor).","hedge":{"p":1.0,"label":"asserted"},"own_p":0.04,"main_claim_p":0.05,"quantitative":true,"quantitative_p":0.8,"role_probabilities":{"gap":0.0,"method":0.0,"result":0.0,"background":1.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":4,"p":1.0,"id":"s4","own":false,"role":"gap","text":"What has been missing so far is a finite-size method for evaluating the quantum-classical guessing-moment separation induced by a given advice distribution.","hedge":{"p":0.99,"label":"asserted"},"own_p":0.06,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.06,"role_probabilities":{"gap":1.0,"method":0.0,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}},{"i":5,"p":0.79,"id":"s5","own":true,"role":"contribution","text":"We provide such a method for product-distribution advice, thereby sharpening the previous entropy-based estimate of Bashiri et al. We reduce the classical and quantum guessing moments to functionals of the one-dimensional surprisal distribution, obtained for product advice by convolving the per-coordinate surprisal laws.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.96,"main_claim_p":0.37,"quantitative":false,"quantitative_p":0.2,"role_probabilities":{"gap":0.0,"method":0.21,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.79,"interpretation":0.0}},{"i":6,"p":0.98,"id":"s6","own":true,"role":"method","text":"When the surprisals lie on a common arithmetic grid (the commensurate case), the logarithmic moments and hence the speedup exponent can be evaluated as finite sums without discretization error; exponential tilting makes this computation numerically stable.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.84,"main_claim_p":0.09,"quantitative":false,"quantitative_p":0.19,"role_probabilities":{"gap":0.0,"method":0.98,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.02,"interpretation":0.0}},{"i":7,"p":0.97,"id":"s7","own":true,"role":"method","text":"For general product advice, we discretize the surprisals onto a common grid and derive an a-posteriori bound on the resulting binning error.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.95,"main_claim_p":0.05,"quantitative":true,"quantitative_p":0.64,"role_probabilities":{"gap":0.0,"method":0.97,"result":0.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.03,"interpretation":0.0}},{"i":8,"p":0.74,"id":"s8","own":true,"role":"contribution","text":"We apply the framework to cold-boot leakage on seeds and block-cipher keys, to template-attack posteriors, and to synthetic i.i.d.","hedge":{"p":0.98,"label":"asserted"},"own_p":0.9,"main_claim_p":0.05,"quantitative":false,"quantitative_p":0.05,"role_probabilities":{"gap":0.0,"method":0.22,"result":0.04,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.74,"interpretation":0.0}},{"i":9,"p":0.55,"id":"s9","own":true,"role":"method","text":"Bernoulli posteriors calibrated to residual ranks reported for Keccak side-channel attacks on ML-KEM and ML-DSA.","hedge":{"p":0.68,"label":"unclear"},"own_p":0.75,"main_claim_p":0.04,"quantitative":false,"quantitative_p":0.08,"role_probabilities":{"gap":0.0,"method":0.55,"result":0.1,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.35,"interpretation":0.0}},{"i":10,"p":1.0,"id":"s10","own":true,"role":"result","text":"The resulting exponents substantially exceed 2 in several skewed-advice settings, reaching up to 3.97 in these synthetic models, and include cases where the previous entropy-based bound did not establish an exponent above 2.","hedge":{"p":1.0,"label":"asserted"},"own_p":0.94,"main_claim_p":0.63,"quantitative":true,"quantitative_p":0.99,"role_probabilities":{"gap":0.0,"method":0.0,"result":1.0,"background":0.0,"limitation":0.0,"future_work":0.0,"contribution":0.0,"interpretation":0.0}}]},"compiled":{"main_claim":{"p":0.86,"text":"The resulting exponents substantially exceed 2 in several skewed-advice settings, reaching up to 3.97 in these synthetic models, and include cases where the previous entropy-based bound did not establish an exponent above 2.","location":"abstract s10","hedge_strength":{"p":1.0,"label":"asserted"}},"provenance":{"route":{"via":"TypeSafe API","model":"classifier-1.13.0","endpoint":"https://api.typesafe.ai/v1/systemone","provider":"TypeSafe"},"packing":"one request per paper","requests":1,"input_tokens":8445,"cost_usd_reported":0.0},"compiled_at":"2026-09-24","bank_version":"abstract-p1-20260923","model_version":"classifier-1.13.0","abstract_states":{"claims_first":0.44,"states_limitation":0.57,"human_participants":0.04,"controlled_comparison":0.46,"randomized_assignment":0.01,"states_quantitative_result":0.98,"compares_with_prior_results":0.97,"mentions_code_or_data_availability":0.03},"evidential_basis":{"p":0.67,"value":"proof or derivation","probabilities":{"argument":0.0,"literature":0.0,"proof_or_derivation":0.67,"analysis_of_existing_data":0.01,"observation_or_experiment":0.0,"simulation_or_computation":0.32}},"contribution_type":{"p":1.0,"value":"method or tool","probabilities":{"review":0.0,"method_or_tool":1.0,"empirical_study":0.0,"simulation_study":0.0,"theoretical_result":0.0,"dataset_or_benchmark":0.0,"position_or_commentary":0.0}},"contribution_statement":{"p":1.0,"text":"We provide such a method for product-distribution advice, thereby sharpening the previous entropy-based estimate of Bashiri et al. 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