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Low-End Uniform Hardness vs. Randomness Tradeoffs for Arthur-Merlin Games. Ronen Shaltiel, University of Haifa Chris Umans, Caltech

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Arthur-Merlin Games [B,GMR] Interactive games in which the all- powerful prover Merlin attempts to prove some statement to a probabilistic poly- time verifier. Merlin Arthur x L toss coins coin tosses message I accept

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Arthur-Merlin Games [B,GMR] Completeness: If the statement is true then Arthur accepts. Soundness: If the statement is false then Pr[Arthur accepts]< ½. Merlin Arthur x L toss coins coin tosses message I accept

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Arthur-Merlin Games [B,GMR] Completeness: If the statement is true then Arthur accepts. Soundness: If the statement is false then Pr[Arthur accepts]< ½. The class AM: All languages L which have an Arthur-Merlin protocol. Contains many interesting problems not known to be in NP.

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Derandomization goals: Efficient deterministic simulation of prob. algs. BPP=P BPP SUBEXP=DTIME(2 n o(1) ) Efficient nondeterministic simulation of prob. protocols AM=NP AM NSUBEXP=NTIME(2 n o(1) ) We don t know how to separate BPP from NEXP. Such a separation implies certain circuit lower bounds [IKW01,KI02].

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Hardness versus Randomness Initiated by [BM,Yao,Shamir,NW]. Assumption: hard functions exist. Conclusion: Derandomization. A lot of works: [BM82,Y82,HILL,NW88,BFNW93, I95,IW97,IW98,KvM99,STV99,ISW99,MV99, ISW00,SU01,U02,TV02,GST03,SU05,U05, … ]

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A quick survey of (nonuniform) hardness/randomness tradeoffs Assumption: There exists a function in E=DTIME(2 O(l) ) which is hard for small (size s(l)) circuits. AMBPP Nondeterministic circuits Deterministic circuits A hard function for: AM=NP [KvM99,MV99,SU05] BPP=P [IW97,STV99] High-end: s(l)=2 Ω(l) AM NSUBEXP [SU01,SU05] BPP SUBEXP [BFNW93,SU01,U02] Low-end: s(l)=l ω(1)

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A quick survey of (uniform) hardness/randomness tradeoffs Assumption: There exists a function in E=DTIME(2 O(l) ) which is hard for small (time s(l)) algorithms/protocols. AMBPP AM protocols.Prob. algsA hard function for: AM=NP (*) [GST03] BPP=P (*) [TV02*] High-end: s(l)=2 Ω(l) AM NSUBEXP (*) BPP SUBEXP (*) [IW98] Low-end: s(l)=l ω(1) This paper* (*) The simulation only succeeds on feasibly generated inputs.

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A low-end gap theorem for AM. Informal statement: Either AM protocols are very strong. Or, AM protocols are somewhat weak. Formal statement: Either E=DTIME(2 O(l) ) has Arthur-Merlin protocols running in time 2 (log l) 3. Or, for every L AM there is a nondeterministic machine M that runs in subexponential time and agrees with L on feasibly generated inputs. No polynomial time machine can produce inputs on which M fails. Should have been poly(l) Jargon: Just like [IW98] paper but for AM instead of BPP

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A uniform hardness vs. randomness tradeoff for AM Informal statement: Either AM protocols are very strong. Or, AM protocols are somewhat weak. Formal statement: For l~~
~~

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Motivation: weak unconditional derandomization We believe that AM=NP (= Σ 1 ). We only know that AM is in Σ 3. Goal: Unconditional proof that AM Σ 2 (or even AM Σ 2 -TIME(2 n o(1) ). Conditional => Unconditional ?? Approach [GST03]: Either AM is weak: AM=NP AM Σ 2. Or AM is very strong: AM=E ??? AM=coAM Σ 2. Missing step: remove feasibly generated inputs.

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A low-end gap theorem for AM coAM. Informal statement: Either AM coAM is very strong. Or, AM coAM is somewhat weak. Formal statement: Either E=DTIME(2 O(l) ) has Arthur-Merlin protocols running in time 2 (log l) 3. Or, for every L AM coAM there is a nondeterministic machine M that runs in subexponential time and agrees with L on all inputs (not necessarily feasibly generated). Should have been poly(l)

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Plan for rest of talk Explain the overall approach of getting uniform hardness vs. randomness tradeoffs for AM (which is in [GST03]). This approach uses a hitting-set generator construction by [MV99] which only works in the high end. Main technical contribution of this paper is improving the [MV99] construction so that it works in the low-end. Improvement uses PCP tools which were not used previously in this framework.

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The high level approach (following [GST03])

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The uniform tradeoff of [GST03]: resilient AM protocols Arthur Merlin nonuniform advice Constructs nondet. Circuit C that is supposed to compute f f(y)=? f(y)=b witness showing C(y)=b AM protocol verifying that C=f. (exists as f is complete for E) C is supposed to define a function: For every y, C is supposed to have witnesses showing C(y)=0 or C(y)=1 but not both! (single valued circuit) Use nonuniform tradeoffs for AM. Derandomization fails => hard function f has small nondeterministic circuits. Want to show that: => f has small AM protocol. Observation [GST03]: The [MV99] tradeoff has an AM protocol in which Arthur verifies that the circuit obtained is single- valued (defines a function). Suppose Arthur could verify that this is indeed the case.

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The uniform tradeoff of [GST03]: Use nonuniform tradeoffs for AM. Derandomization fails => hard function f has small nondeterministic circuits. Want to show that: => f has small AM protocol. Observation [GST03]: The [MV99] tradeoff has an AM protocol in which Arthur verifies that the circuit obtained is single- valued (defines a function). => AM protocol for f. Problem: The [MV99] generator only works in the high end. Our contribution: Modify [MV99] into a low-end generator. Arthur Merlin C is supposed to compute f f(y)=? f(y)=b witness showing C(y)=b AM protocol verifying that C=f. (exists as f is complete for E) AM protocol in which Arthur receives a certified valid circuit C C is guaranteed to define a function: For every y, C is has witnesses showing C(y)=0 or C(y)=1 but not both!

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Abstraction: commit-and-evaluate AM protocols and resiliency. Commit-and-evaluate AM protocols for function f(y). Properties: Input y can be revealed to Merlin after commit phase. Conformity: Honest Merlin can make Arthur output f(y). Resiliency: Following commit phase Merlin is (w.h.p) committed to some function g(y) (may differ from f). Thm: If E has such AM protocols then E has standard AM protocols. Arthur Merlin f(y)=? f(y)=b Evaluation phase: AM protocol that uses advice string and outputs a value v(y). Commit phase: AM protocol generating advice string.

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The big picture: How to derandomize an AM protocol Nondet machine M(x) (supposed to accept L AM) Use function f to construct small hitting set of pseudorandom strings. Run AM protocol on input x (using pseudorandom strings as random coins) and accept if all runs accept. Proof of correctness by reduction Suppose M fails on an input x. Construct an efficient commit- and-evaluate AM protocol that uses x and conforms resiliently with f. => f has a standard efficient AM protocol. Where do feasibly generated inputs come in? How can Arthur obtain x? From his point of view x is a nonuniform advice string. No problem if we only care about inputs that can be feasibly generated by some efficient TM. Following [GST03]: In the case of AM coAM we can trust Merlin to send a good x. This is where uniformity comes in: Protocols rather than circuits.

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Improving Miltersen-Vinodchandran hitting set generator (How to derandomize an AM language)

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The MV hitting set generator Nondet. Machine M(x): derandomizes AM protocol with m coins For every output string guess a response for Merlin and accept if Arthur accepts all of them. x L Merlin can answer any string M accepts (no error). xL Merlin can answer ½ strings M may err. truth table f:{0,1} l {0,1} field size q=102 l/2 p(x,y) deg. 2 l/2 A {0,1} m extractor rows and columns Z small set (2 m ) deg 2 l/2 polys Hitting set

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A commit-and-evaluate AM protocol for p. truth table f:{0,1} l {0,1} field size q=102 l/2 p(x,y) deg. 2 l/2 A {0,1} m extractor rows and columns Z very small set

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truth table f:{0,1} l {0,1} field size q=102 l/2 p(x,y) deg. 2 l/2 A {0,1} m extractor rows and columns Z small set (2 m ) A commit-and-evaluate AM protocol for p. Conformity: v Resiliency: v (RS code). w.h.p. over S univariate poly g Z, g(S) is unique. Efficiency: (on high end m=2 Ω(l) ). Protocol runs in time poly(m). Protocol requires passing polynomials of degree 2 l/2. Arthur Merlin p(x,y)=? p(x,y )= b commit phase: S R Field of size m p(S 2 ) evaluation phase: Both compute path to (x,y) line on path: p| line and witness w Arthur checks: small set: p| line Z using w. consistency of p| line. deg 2 l/2 polys

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truth table f:{0,1} l {0,1} field size q=102 l/2 p(x,y) deg. 2 l/2 A {0,1} m extractor rows and columns Z very small set A commit-and-evaluate AM protocol for p. Conformity: v Resiliency: v (RS code). w.h.p. over S univariate poly gZ, g(S) is unique. Efficiency: (on high end m=2 Ω(l) ). Protocol runs in time poly(m). Protocol requires passing polynomials of degree 2 l/2. Arthur Merlin commit phase: S R Field of size m p(S 2 ) evaluation phase: Both compute path to (x,y) line on path: p| line and witness w Arthur checks: small set: p| line Z using w. consistency of p| line. For low end (say m=l O(1) ) we need to reduce degree Can get deg=2 l/d using p(x 1,..,x d ) with d variables. S 2 S d |S d |=2 Ω(l) no gain! p(x,y)=? p(x,y )= b

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truth table f:{0,1} l {0,1} field size q=102 l/2 p(x,y) deg. 2 l/2 A {0,1} m extractor rows and columns Z very small set The best of both worlds Best of both worlds: Use p(x 1,..,x d ) deg 2 l/d. Run MV as if p=p(x,y). Resiliency: v (RM code). Size of box=|S| 2 m 2. doesnt depend on d! Sending p| line costs 2 l/2 bits. Arthur Merlin commit phase: S R Field of size m p(S 2 ) evaluation phase: Both compute path to (x,y) line on path: p| line and witness w Arthur checks: small set: p| line Z using w. consistency of p| line. deg 2 l/2 polys p(x,y)=? p(x,y )= b p(x 1,..,x d/2 ;x d/2+1,..,x d ). p| line has many coefficients Suppose p| line could be sent more efficiently: p| line has small (non- det) circuit p| line has commit-and- evaluate protocol Arthur can verify that he gets a low-degree polynomial by performing low-degree testing!

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truth table f:{0,1} l {0,1} field size q=102 l/2 p(x,y) deg. 2 l/2 A {0,1} m extractor rows and columns Z very small set Locally computable extractors Story so far: Use polynomials p with many variables and pretend they are bivariate. Assume that p| line can be sent efficiently. (Not yet justified). Is the AM protocol efficient? Arthur Merlin commit phase: S R Field of size m p(S 2 ) evaluation phase: Both compute path to (x,y) line on path: p| line and witness w Arthur checks: small set: p| line Z using w. consistency of p| line. deg 2 l/2 polys p(x,y)=? p(x,y )= b v v v v v X Requires running the extractor on p| line (t) for all q d inputs t to p| line. Need locally computable extractors! Thm: [V] no locally computable extractors for low-entropy. We know that inputs to extractors are low-degree polynomials! Can use extractor construction [SU01] which is locally computable. Thm: [V] no locally computable extractors for low-entropy. We know that inputs to extractors are low-degree polynomials! Can use extractor construction [SU01] which is locally computable. v Efficient AM protocol!

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v truth table f:{0,1} l {0,1} field size q=102 l/2 p(x,y) deg. 2 l/2 A {0,1} m extractor rows and columns Z very small set Win-win analysis Main ideas: Use polynomials p with many variables and pretend they are bivariate. Assume that p| line can be sent efficiently. (Not yet justified). Use locally computable extractors (exist when inputs are low degree polynomials. Arthur Merlin commit phase: S R Field of size m p(S 2 ) evaluation phase: Both compute path to (x,y) line on path: p| line and witness w Arthur checks: small set: p| line Z using w. consistency of p| line. deg 2 l/2 polys p(x,y)=? p(x,y )= b v v v v v Efficient AM protocol! ???? Intuition: If p| line doesnt have an efficient commit-and-evaluate protocol then its better to use it as the hard function. (It is over less variables!) Recursive win-win analysis a-la [ISW99].

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The recursive HSG truth table of f … each row and column ALL rows and columns to extractor (2 l bits) (2 l/2 bits each )

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Recursive commit-and-evaluate AM protocol truth table of f … Arthur: random m x m box Merlin: commits to top board ! (input revealed) Arthur/ Merlin: commit to each lines board Arthur: random points for checking lines Merlin: commits to lines…

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Parameters Start with 2 l bit truth table < log l levels # lines 2 O(l). v Efficiency of AM protocol: poly(m) blow-up at each level ) poly(m) log l running time convert O(log l) rounds to two rounds ) poly(m) (log l) 2 time for final AM protocol

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Conclusions Key ideas: commit-and-evaluate protocols as abstraction. operate implicitly on lines. PCP tools: low-degree testing, self-correction. local extractor when know in advance it will applied to low-degree polynomials. Recursive win-win analysis allows large objects to have short descriptions. Open problems: improve poly(m) (log l) 2 to poly(m) ( optimal ). remove feasibly generated inputs from main theorem. Uncondtional proof that AM Σ 2 (TIME(2 n o(1) )).

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That s it …

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