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**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

Motivated by the quest for scalable and succinct zero knowledge arguments, we revisit worst-case-to-average-case reductions for linear spaces, raised by [Rothblum, Vadhan, Wigderson, STOC 2013]. The previous state of the art by [Ben-Sasson, Kopparty, Saraf, CCC 2018] showed that if some member of an affine space U is δ-far in relative Hamming distance from a linear code V - this is the worst-case assumption - then most elements of U are almost-δ-far from V - this is the average case. However, this result was known to hold only below the "double Johnson" function of the relative distance δ_V of the code V, i.e., only when δ < 1-(1-δ_V)^(1/4).
First, we increase the soundness-bound to the "one-and-a-half Johnson" function of δ_V and show that the average distance of U from V is nearly δ for any worst-case distance δ smaller than 1-(1-δ_V)^(1/3). This bound is tight, which is somewhat surprising because the one-and-a-half Johnson function is unfamiliar in the literature on error correcting codes.
To improve soundness further for Reed Solomon codes we sample outside the box. We suggest a new protocol in which the verifier samples a single point z outside the box D on which codewords are evaluated, and asks the prover for the value at z of the interpolating polynomial of a random element of U. Intuitively, the answer provided by the prover "forces" it to choose one codeword from a list of "pretenders" that are close to U. We call this technique Domain Extending for Eliminating Pretenders (DEEP).
The DEEP method improves the soundness of the worst-case-to-average-case reduction for RS codes up their list decoding radius. This radius is bounded from below by the Johnson bound, implying average distance is approximately δ for all δ < 1-(1-δ_V)^(1/2). Under a plausible conjecture about the list decoding radius of Reed-Solomon codes, average distance from V is approximately δ for all δ. The DEEP technique can be generalized to all linear codes, giving improved reductions for capacity-achieving list-decodable codes.
Finally, we use the DEEP technique to devise two new protocols:
- An Interactive Oracle Proof of Proximity (IOPP) for RS codes, called DEEP-FRI. The soundness of the protocol improves upon that of the FRI protocol of [Ben-Sasson et al., ICALP 2018] while retaining linear arithmetic proving complexity and logarithmic verifier arithmetic complexity.
- An Interactive Oracle Proof (IOP) for the Algebraic Linking IOP (ALI) protocol used to construct zero knowledge scalable transparent arguments of knowledge (ZK-STARKs) in [Ben-Sasson et al., eprint 2018]. The new protocol, called DEEP-ALI, improves soundness of this crucial step from a small constant < 1/8 to a constant arbitrarily close to 1.

Eli Ben-Sasson, Lior Goldberg, Swastik Kopparty, and Shubhangi Saraf. DEEP-FRI: Sampling Outside the Box Improves Soundness. In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 5:1-5:32, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bensasson_et_al:LIPIcs.ITCS.2020.5, author = {Ben-Sasson, Eli and Goldberg, Lior and Kopparty, Swastik and Saraf, Shubhangi}, title = {{DEEP-FRI: Sampling Outside the Box Improves Soundness}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {5:1--5:32}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.5}, URN = {urn:nbn:de:0030-drops-116901}, doi = {10.4230/LIPIcs.ITCS.2020.5}, annote = {Keywords: Interactive Oracle Proofs of Proximity, STARK, Low Degree Testing} }

Document

**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

This paper studies families of distributions T that are amenable to retentive learning, meaning that an expert can retain users that seek to predict their future, assuming user attributes are sampled from T and exposed gradually over time. Limited attention span is the main problem experts face in our model. We make two contributions.
First, we formally define the notions of retentively learnable distributions and properties. Along the way, we define a retention complexity measure of distributions and a natural class of retentive scoring rules that model the way users evaluate experts they interact with. These rules are shown to be tightly connected to truth-eliciting "proper scoring rules" studied in Decision Theory since the 1950's [McCarthy, PNAS 1956].
Second, we take a first step towards relating retention complexity to other measures of significance in computational complexity. In particular, we show that linear properties (over the binary field) are retentively learnable, whereas random Low Density Parity Check (LDPC) codes have, with high probability, maximal retention complexity. Intriguingly, these results resemble known results from the field of property testing and suggest that deeper connections between retentive distributions and locally testable properties may exist.

Eli Ben-Sasson and Eden Saig. The Complexity of User Retention. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 12:1-12:30, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bensasson_et_al:LIPIcs.ITCS.2019.12, author = {Ben-Sasson, Eli and Saig, Eden}, title = {{The Complexity of User Retention}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {12:1--12:30}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.12}, URN = {urn:nbn:de:0030-drops-101053}, doi = {10.4230/LIPIcs.ITCS.2019.12}, annote = {Keywords: retentive learning, retention complexity, information elicitation, proper scoring rules} }

Document

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

The family of Reed-Solomon (RS) codes plays a prominent role in the construction of quasilinear probabilistically checkable proofs (PCPs) and interactive oracle proofs (IOPs) with perfect zero knowledge and polylogarithmic verifiers. The large concrete computational complexity required to prove membership in RS codes is one of the biggest obstacles to deploying such PCP/IOP systems in practice.
To advance on this problem we present a new interactive oracle proof of proximity (IOPP) for RS codes; we call it the Fast RS IOPP (FRI) because (i) it resembles the ubiquitous Fast Fourier Transform (FFT) and (ii) the arithmetic complexity of its prover is strictly linear and that of the verifier is strictly logarithmic (in comparison, FFT arithmetic complexity is quasi-linear but not strictly linear). Prior RS IOPPs and PCPs of proximity (PCPPs) required super-linear proving time even for polynomially large query complexity.
For codes of block-length N, the arithmetic complexity of the (interactive) FRI prover is less than 6 * N, while the (interactive) FRI verifier has arithmetic complexity <= 21 * log N, query complexity 2 * log N and constant soundness - words that are delta-far from the code are rejected with probability min{delta * (1-o(1)),delta_0} where delta_0 is a positive constant that depends mainly on the code rate. The particular combination of query complexity and soundness obtained by FRI is better than that of the quasilinear PCPP of [Ben-Sasson and Sudan, SICOMP 2008], even with the tighter soundness analysis of [Ben-Sasson et al., STOC 2013; ECCC 2016]; consequently, FRI is likely to facilitate better concretely efficient zero knowledge proof and argument systems.
Previous concretely efficient PCPPs and IOPPs suffered a constant multiplicative factor loss in soundness with each round of "proof composition" and thus used at most O(log log N) rounds. We show that when delta is smaller than the unique decoding radius of the code, FRI suffers only a negligible additive loss in soundness. This observation allows us to increase the number of "proof composition" rounds to Theta(log N) and thereby reduce prover and verifier running time for fixed soundness.

Eli Ben-Sasson, Iddo Bentov, Yinon Horesh, and Michael Riabzev. Fast Reed-Solomon Interactive Oracle Proofs of Proximity. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bensasson_et_al:LIPIcs.ICALP.2018.14, author = {Ben-Sasson, Eli and Bentov, Iddo and Horesh, Yinon and Riabzev, Michael}, title = {{Fast Reed-Solomon Interactive Oracle Proofs of Proximity}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {14:1--14:17}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.14}, URN = {urn:nbn:de:0030-drops-90183}, doi = {10.4230/LIPIcs.ICALP.2018.14}, annote = {Keywords: Interactive proofs, low degree testing, Reed Solomon codes, proximity testing} }

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Brief Announcement

**Published in:** LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

A theoretical model is suggested for abstracting the interaction between an expert system and its users, with a focus on reputation and incentive compatibility. The model assumes users interact with the system while keeping in mind a single "retention parameter" that measures the strength of their belief in its predictive power, and the system's objective is to reinforce and maximize this parameter through "informative" and "correct" predictions.
We define a natural class of retentive scoring rules to model the way users update their retention parameter and thus evaluate the experts they interact with. Assuming agents in the model have an incentive to report their true belief, these rules are shown to be tightly connected to truth-eliciting "proper scoring rules" studied in Decision Theory.
The difference between users and experts is modeled by imposing different limits on their predictive abilities, characterized by a parameter called memory span. We prove the monotonicity theorem ("more knowledge is better"), which shows that experts with larger memory span retain better in expectation.
Finally, we focus on the intrinsic properties of phenomena that are amenable to collaborative discovery with a an expert system. Assuming user types (or "identities") are sampled from a distribution D, the retention complexity of D is the minimal initial retention value (or "strength of faith") that a user must have before approaching the expert, in order for the expert to retain that user throughout the collaborative discovery, during which the user "discovers" his true "identity". We then take a first step towards relating retention complexity to other established computational complexity measures by studying retention dynamics when D is a uniform distribution over a linear space.

Eli Ben-Sasson and Eden Saig. Brief Announcement: Towards an Abstract Model of User Retention Dynamics. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 164:1-164:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bensasson_et_al:LIPIcs.ICALP.2018.164, author = {Ben-Sasson, Eli and Saig, Eden}, title = {{Brief Announcement: Towards an Abstract Model of User Retention Dynamics}}, booktitle = {45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)}, pages = {164:1--164:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-076-7}, ISSN = {1868-8969}, year = {2018}, volume = {107}, editor = {Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.164}, URN = {urn:nbn:de:0030-drops-91682}, doi = {10.4230/LIPIcs.ICALP.2018.164}, annote = {Keywords: information elicitation, proper scoring rules, retention complexity} }

Document

**Published in:** LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

Algebraic proof systems reduce computational problems to problems about estimating the distance of a sequence of functions vec{u}=(u_1,..., u_k), given as oracles, from a linear error correcting code V. The soundness of such systems relies on methods that act "locally" on vec{u} and map it to a single function u^* that is, roughly, as far from V as are u_1,..., u_k.
Motivated by these applications to efficient proof systems, we study a natural worst-case to average-case reduction of distance for linear spaces, and show several general cases in which the following statement holds: If some member of a linear space U=span(u_1,...,u_k) is delta-far from (all elements) of V in relative Hamming distance, then nearly all elements of U are (1-epsilon)delta-far from V; the value of epsilon depends only on the distance of the code V and approaches 0 as that distance approaches 1. Our results improve on the previous state-of-the-art which showed that nearly all elements of U are 1/2delta-far from V [Rothblum, Vadhan and Wigderson, STOC 2013].
When V is a Reed-Solomon (RS) code, as is often the case for algebraic proof systems, we show how to boost distance via a new "local" transformation that may be useful elsewhere. Relying on the affine-invariance of V, we map a vector u to a random linear combination of affine transformations of u, and show this process amplifies distance from V. Assuming V is an RS code with sufficiently large distance, this amplification process converts a function u that is somewhat far from V to one that is (1-epsilon)-far from V; as above, epsilon depends only on the distance of V and approaches 0 as the distance of V approaches 1.
We give two concrete application of these techniques. First, we revisit the axis-parallel low-degree test for bivariate polynomials of [Polischuk-Spielman, STOC 1994] and prove a "list-decoding" type result for it, when the degree of one axis is extremely small. This result is similar to the recent list-decoding-regime result of [Chiesa, Manohar and Shinkar, RANDOM 2017] but is proved using different techniques, and allows the degree in one axis to be arbitrarily large. Second, we improve the soundness analysis of the recent RS proximity testing protocol of [Ben-Sasson et al., ICALP 2018] and extend it to the "list-decoding" regime, bringing it closer to the Johnson bound.

Eli Ben-Sasson, Swastik Kopparty, and Shubhangi Saraf. Worst-Case to Average Case Reductions for the Distance to a Code. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 24:1-24:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bensasson_et_al:LIPIcs.CCC.2018.24, author = {Ben-Sasson, Eli and Kopparty, Swastik and Saraf, Shubhangi}, title = {{Worst-Case to Average Case Reductions for the Distance to a Code}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {24:1--24:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.24}, URN = {urn:nbn:de:0030-drops-88654}, doi = {10.4230/LIPIcs.CCC.2018.24}, annote = {Keywords: Proximity testing, Reed-Solomon codes, algebraic coding complexity} }

Document

**Published in:** LIPIcs, Volume 80, 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)

We study interactive oracle proofs (IOPs) [BCS16,RRR16], which combine aspects of probabilistically checkable proofs (PCPs) and interactive proofs (IPs). We present IOP constructions and techniques that enable us to obtain tradeoffs in proof length versus query complexity that are not known to be achievable via PCPs or IPs alone. Our main results are:
1. Circuit satisfiability has 3-round IOPs with linear proof length (counted in bits) and constant query complexity.
2. Reed-Solomon codes have 2-round IOPs of proximity with linear proof length and constant query complexity.
3. Tensor product codes have 1-round IOPs of proximity with sublinear proof length and constant query complexity.
For all the above, known PCP constructions give quasilinear proof length and constant query complexity [BS08,Din07]. Also, for circuit satisfiability, [BKKMS13] obtain PCPs with linear proof length but sublinear (and super-constant) query complexity. As in [BKKMS13], we rely on algebraic-geometry codes to obtain our first result; but, unlike that work, our use of such codes is much "lighter" because we do not rely on any automorphisms of the code.
We obtain our results by proving and combining "IOP-analogues" of tools underlying numerous IPs and PCPs:
* Interactive proof composition. Proof composition [AS98] is used to reduce the query complexity of PCP verifiers, at the cost of increasing proof length by an additive factor that is exponential in the verifier's randomness complexity. We prove a composition theorem for IOPs where this additive factor is linear.
* Sublinear sumcheck. The sumcheck protocol [LFKN92] is an IP that enables the verifier to check the sum of values of a low-degree multi-variate polynomial on an exponentially-large hypercube, but the verifier's running time depends linearly on the bound on individual degrees. We prove a sumcheck protocol for IOPs where this dependence is sublinear (e.g., polylogarithmic).
Our work demonstrates that even constant-round IOPs are more efficient than known PCPs and IPs.

Eli Ben-Sasson, Alessandro Chiesa, Ariel Gabizon, Michael Riabzev, and Nicholas Spooner. Interactive Oracle Proofs with Constant Rate and Query Complexity. In 44th International Colloquium on Automata, Languages, and Programming (ICALP 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 80, pp. 40:1-40:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{bensasson_et_al:LIPIcs.ICALP.2017.40, author = {Ben-Sasson, Eli and Chiesa, Alessandro and Gabizon, Ariel and Riabzev, Michael and Spooner, Nicholas}, title = {{Interactive Oracle Proofs with Constant Rate and Query Complexity}}, booktitle = {44th International Colloquium on Automata, Languages, and Programming (ICALP 2017)}, pages = {40:1--40:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-041-5}, ISSN = {1868-8969}, year = {2017}, volume = {80}, editor = {Chatzigiannakis, Ioannis and Indyk, Piotr and Kuhn, Fabian and Muscholl, Anca}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2017.40}, URN = {urn:nbn:de:0030-drops-74713}, doi = {10.4230/LIPIcs.ICALP.2017.40}, annote = {Keywords: probabilistically checkable proofs, interactive proofs, proof composition, sumcheck} }

Document

**Published in:** Dagstuhl Seminar Proceedings, Volume 8381, Computational Complexity of Discrete Problems (2008)

We continue the study of tradeoffs between space and length of
resolution proofs and focus on two new results:
begin{enumerate}
item
We show that length and space in resolution are uncorrelated. This
is proved by exhibiting families of CNF formulas of size $O(n)$ that
have proofs of length $O(n)$ but require space $Omega(n / log n)$. Our
separation is the strongest possible since any proof of length $O(n)$
can always be transformed into a proof in space $O(n / log n)$, and
improves previous work reported in [Nordstr"{o}m 2006, Nordstr"{o}m and
H{aa}stad 2008].
item We prove a number of trade-off results for space in the range
from constant to $O(n / log n)$, most of them superpolynomial or even
exponential. This is a dramatic improvement over previous results in
[Ben-Sasson 2002, Hertel and Pitassi 2007, Nordstr"{o}m 2007].
end{enumerate}
The key to our results is the following, somewhat surprising, theorem:
Any CNF formula $F$ can be transformed by simple substitution
transformation into a new formula $F'$ such that if $F$ has the right
properties, $F'$ can be proven in resolution in essentially the same
length as $F$ but the minimal space needed for $F'$ is lower-bounded
by the number of variables that have to be mentioned simultaneously in
any proof for $F$. Applying this theorem to so-called pebbling
formulas defined in terms of pebble games over directed acyclic graphs
and analyzing black-white pebbling on these graphs yields our results.

Eli Ben-Sasson and Jakob Nordström. Understanding space in resolution: optimal lower bounds and exponential trade-offs. In Computational Complexity of Discrete Problems. Dagstuhl Seminar Proceedings, Volume 8381, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2008)

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@InProceedings{bensasson_et_al:DagSemProc.08381.6, author = {Ben-Sasson, Eli and Nordstr\"{o}m, Jakob}, title = {{Understanding space in resolution: optimal lower bounds and exponential trade-offs}}, booktitle = {Computational Complexity of Discrete Problems}, series = {Dagstuhl Seminar Proceedings (DagSemProc)}, ISSN = {1862-4405}, year = {2008}, volume = {8381}, editor = {Peter Bro Miltersen and R\"{u}diger Reischuk and Georg Schnitger and Dieter van Melkebeek}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/DagSemProc.08381.6}, URN = {urn:nbn:de:0030-drops-17815}, doi = {10.4230/DagSemProc.08381.6}, annote = {Keywords: Proof complexity, Resolution, Pebbling.} }

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