[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-85427-en":3,"doc-seo-85427-105":29,"detail-sidebar-cat-0-en-105":91},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":13,"seo_description":14,"update_tm":27,"read_time":28},85427,687197100911,"Himbo","https://ap-avatar.wpscdn.com/avatar/a000239b6f1da00475?x-image-process=image/resize,m_fixed,w_180,h_180&k=1782698725881665579",8,"Research & Report","Generalized and Unified Equivalences between Hardness and Pseudoentropy","Pseudoentropy characterizations establish quantitative links between computational hardness and computational randomness. The work proves a unified pseudoentropy characterization that generalizes and strengthens earlier results in both uniform and nonuniform computation models. The characterization spans a broad family of entropy notions, including Shannon entropy and min-entropy, and is simultaneously witnessed by a single universal function reflecting both hardness and randomness. Weight-restricted calibration combined with multiaccuracy motivates an enhanced regularity and leakage simulation lemma, improving alphabet-size dependence.","Generalized and Unified Equivalences between Hardness and Pseudoentropy  \nLunjia Hu∗  \nNortheastern University [lunjia@alumni. stanford. edu](lunjia@alumni. stanford. edu)  \nSalil Vadhan†  \nHarvard University [salil_vadhan@harvard. edu](salil_vadhan@harvard. edu)  \narXiv :2507 .05972v 3 [ cs .CC] 10 Jul 2026  \nAbstract  \nPseudoentropy characterizations give quantitatively precise formulations of the relationship between computational hardness and computational randomness. We prove a unified pseudoentropy characterization that generalizes and strengthens previous results in both uniform and nonuniform models of computation. Our characterization applies to a general family of entropy notions, including Shannon entropy and min-entropy as special cases. Moreover, the characterizations for these different entropy notions can be witnessed simultaneously by a single universal function, which captures both computational hardness and computational randomness.  \nA key technical insight is that weight-restricted calibration, from the recent literature on algorithmic fairness, together with standard computational indistinguishability (known as multiaccuracy in the fairness literature), suffices for proving pseudoentropy characterizations for general entropy notions. To obtain this combination of properties, we prove an enhanced version of the Leakage Simulation Lemma (Jetchev and Pietrzak, 2014), which in turn extends the Complexity Theoretic-Regularity Lemma (Trevisan, Tulsiani, and Vadhan, 2009) from boolean functions to ones over a larger alphabet. Our Enhanced Regularity/Leakage-Simulation Lemma enables us to obtain an exponential improvement in the dependence on the alphabet size compared with the pseudoentropy characterizations of Casacuberta, Dwork, and Vadhan (2024), which are based on the stronger notion of multicalibration. We also show that this exponential dependence on the alphabet size is inevitable for multicalibration and even for the weaker notion of calibrated multiaccuracy.  \nOur Enhanced Regularity/Leakage-Simulation Lemma can be viewed as an extension of a result about “sample-access outcome indistinguishability” (Dwork, Kim, Reingold, Rothblum, and Yona, 2021) from boolean functions to ones over a larger alphabet. We hypothesize that these lemmas will prove to be a powerful tool for other applications in average-case complexity and cryptography.  \n1 Introduction  \n1.1 Background on Hardness–Randomness Equivalences  \nThe close relationship between computational hardness and computational randomness is central in cryptography and complexity theory. A classic example of this relationship is Yao’s equivalence  \n∗ Supported by the Simons Foundation Collaboration on the Theory of Algorithmic Fairness and the Harvard Center for Research on Computation and Society. Part of this work was performed while LH was a Postdoctoral Fellow at Harvard University.  \n†Supported by a Simons Investigators Award.  \nbetween pseudorandomness and (maximal) unpredictability [Yao82], one form of which is the following:  \nTheorem 1.1 . Let (X, Y ) be a random variable distributed on {0, 1}n × {0, 1}ℓ with ℓ = O (log n) . Then the following are equivalent:  \n1. (X, Y ) ≈c (X, Uℓ), where ≈c denotes computational indistinguishability (against nonuniform polynomial-time algorithms) .  \n2. For every nonuniform polynomial-time algorithm A,  \n 1  \nPr[A(X) = Y ] ≤ + negl(n)  \n2ℓ .  \nWe refer to Item 2 as maximal unpredictability, because it is trivial for an efficient A to achieve prediction probability 1/2ℓ , by just outputting a uniformly random ℓ-bit string. It is natural to ask what happens if we weaken the hardness of prediction to allow A to succeed with some probability between 1/2ℓ and 1 . Vadhan and Zheng [VZ12 , Zhe14] showed that such weak unpredictability is equivalent to Y having high pseudo-average-min-entropy [HLR07] given X .  \nTheorem 1.2 . Let (X, Y ) be a random variable distributed on {0, 1}n × {0, 1}ℓ with ℓ = O (log n), and let k ∈","cbCairjxuwJmOJwr","https://ap.wps.com/l/cbCairjxuwJmOJwr","pdf",902730,1,47,"English","en",105,"# Abstract\n## Introduction\n### Background on Hardness–Randomness Equivalences","[{\"question\":\"What does the paper mean by a unified pseudoentropy characterization?\",\"answer\":\"It provides one characterization that generalizes and strengthens previous pseudoentropy results across uniform and nonuniform models, tying computational hardness to computational randomness through entropy measures.\"},{\"question\":\"Which entropy notions are covered by the main characterization?\",\"answer\":\"The characterization applies to a general family of entropy notions, with Shannon entropy and min-entropy as special cases.\"},{\"question\":\"What technical tool enables the improved results?\",\"answer\":\"The paper uses weight-restricted calibration together with standard computational indistinguishability (multiaccuracy), achieved via an enhanced Leakage Simulation Lemma and an extension of a regularity lemma to larger 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does the paper mean by a unified pseudoentropy characterization?","Question",{"text":75,"@type":76},"It provides one characterization that generalizes and strengthens previous pseudoentropy results across uniform and nonuniform models, tying computational hardness to computational randomness through entropy measures.","Answer",{"name":78,"@type":73,"acceptedAnswer":79},"Which entropy notions are covered by the main characterization?",{"text":80,"@type":76},"The characterization applies to a general family of entropy notions, with Shannon entropy and min-entropy as special cases.",{"name":82,"@type":73,"acceptedAnswer":83},"What technical tool enables the improved results?",{"text":84,"@type":76},"The paper uses weight-restricted calibration together with standard computational indistinguishability (multiaccuracy), achieved via an enhanced Leakage Simulation Lemma and an extension of a regularity lemma to larger 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