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Extracting Randomness David Zuckerman University of Texas at Austin

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Randomness extremely useful Algorithms –Approximation, optimization, factoring polys. Monte Carlo simulations Cryptography Distributed computing –Consensus, Byzantine agreement, load balancing.

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Randomness wonderful, but … Computers typically don’t have access to true randomness.

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Is Randomness Necessary? Essential for distributed computing and cryptography: –Must choose secret key randomly. Unclear for algorithms.

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Is Randomness Necessary? Major open question in field: does every efficient randomized algorithm have an efficient deterministic counterpart? –Does RP = P?

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Is Randomness Necessary? Major open question in field: does every efficient randomized algorithm have an efficient deterministic counterpart? –Does RP = P? Appears very difficult. –Does RSPACE(S) = SPACE(S)? Difficult but some hope.

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What is minimal randomness requirement? Can we eliminate randomness completely? If not: –Can we minimize quantity of randomness? –Can we minimize quality of randomness? What does this mean?

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What is minimal randomness requirement? Can we eliminate randomness completely? If not: –Can we minimize quantity of randomness? Pseudorandom generator –Can we minimize quality of randomness? Extractor

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Pseudorandom Generators Computers rely on pseudorandom generators: PRG short random string long “random-enough” string Classical approach: ad hoc. Many failures. Modern approach: provably good PRGs.

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Quality: von Neumann’s model Ext very long weakly random long random Bits independent. Each bit has same bias: –Pr[X i =1] = p, p unknown. Can’t use directly. Goal:

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Quality: von Neumann’s model Extractor: –Group bits in pairs. –Pr[01]=Pr[10] = p(1-p). –Map 01 to 0, 10 to 1, ignore 00 and 11. Example: maps to

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Use in Practice Intel has random number generator (not PRG) which uses white noise. Temperature may influence bias. Intel applies von Neumann’s extractor to output.

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General Weakly Random Sources What if bits are correlated? Many models studied [Blum, Santha-Vazirani, Chor-Goldreich]. Most general model - upper bound probability of each string [Zuckerman]. Similar to lower bounding entropy.

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General Weakly Random Sources Weakly random distribution on n bits: each string has probability ≤ 2 -k. Example: weakly random integer in [1,1000]. Distribution unknown.

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Goal Ext very long weakly random long almost random Should work for all (n,k) weakly random sources.

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Goal Ext very long weakly random long almost random Should work for all (n,k) weakly random sources. Problem: impossible.

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Solution: Extractor [Nisan-Zuckerman] Ext very long weakly random long almost random short truly random

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Extractor Parameters [NZ,…, Lu-Reingold-Vadhan-Wigderson] Ext n bits weakly random Pr[each string] ≤ 2 -k.99k bits almost random O(log n) truly random

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Power of Extractors Sometimes can eliminate true randomness by cycling over all possibilities.

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Power of Extractors Sometimes can eliminate true randomness by cycling over all possibilities. Useful even when no weakly random source apparently present.

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Power of Extractors Sometimes can eliminate true randomness by cycling over all possibilities. Useful even when no weakly random source apparently present. Mathematical reason for power: extractor constructions beat “eigenvalue bound.”

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Applications of Extractors PRGs for Space-Bounded Computation [Nisan-Z] PRGs for Random Sampling [Z] Cryptography [Lu, Vadhan, Dodis-Smith] Expander graphs and highly connected networks [Wigderson-Z] Coding theory [Ta-Shma- Z] Hardness of approximation [Z, Mossel-Umans] Efficient deterministic sorting [Pippenger] Time-space tradeoffs [Sipser] Implicit data structures [Fiat-Naor, Z]

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New Extractor and Application [Z] Extractor requires log n + O(1) random bits. NP-complete to approximate MAX CLIQUE and CHROMATIC NUMBER to within n 1- , any >0. –Previously same inapproximability ratio required NP ZPP [Hastad, Feige-Kilian]. –We use new extractor to derandomize previous reductions.

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The Future for Extractors Current extractors near optimal. Where to go from here? Two interesting directions: –Deterministic extractors for specialized sources. –Extractors for independent sources and a new technique.

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Bit-Fixing Sources Adversary fixes all but k of the n bits. Remaining k bits chosen randomly. Parity can extract 1 bit if k≥1.

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Bit-Fixing Sources Adversary fixes all but k of the n bits. Remaining k bits chosen randomly. Parity can extract 1 bit if k≥1. This model seems unrealistic: –What good is it?

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Bit-Fixing Sources Adversary fixes all but k of the n bits. Remaining k bits chosen randomly. Parity can extract 1 bit if k≥1. This model seems unrealistic: –What good is it? Applications in cryptography and more realistic models.

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Bit-Fixing Sources Adversary fixes all but k of the n bits. Remaining k bits chosen randomly. Parity can extract 1 bit if k≥1. To extract 2 truly random bits, need k>n/3. Can extract k 2 /n almost-random bits deterministically [Kamp-Zuckerman]. Improved to (1-o(1))k [Gabizon-Raz-Shaltiel].

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Exposure-Resilient Cryptography Standard cryptography: secret keys totally secret. What if adversary learns some bits of secret key? Deterministic extractors for bit-fixing sources can help foil such adversaries [Dodis-Sahai-Smith]. Need exponentially small error. Kamp-Z extractor has small enough error to apply ([GRS] error too large).

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More realistic sources: Generalizing von Neumann’s Model Bits independent, allow different biases. Deterministic extractors for bit-fixing sources also work for these new sources [Kamp-Vadhan-Zuckerman]. Goal: deterministic extractors for more general sources. Some preliminary results allowing correlations.

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Technique: Additive Number Theory For set A, A+A={a 1 +a 2 : a 1, a 2 in A} Thm: either |A+A|>|A| 1.01 or |A A| > |A| 1.01 [Bourgain-Katz-Tao, Konyagin]. Can extract from 3 independent sources [Barak-Kindler-Shaltiel-Sudakov-Wigerson]. Promising technique -- other applications? Anup Rao: improvements without additive number theory.

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Conclusions Extractors fundamental: diverse applications. Future in extractors: –Deterministic extractors –2-source extractors –More applications –Practical variants Can we make progress towards RP=P or RSPACE(S) = SPACE(S)?

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Students Jesse Kamp - extractors Anindya Patthak - coding theory Anup Rao - extractors

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Extractors in Cryptography Alice and Bob know s = “secret” random 1000 bit string. Eavesdropper Eve knows 600 bits of s. Alice and Bob don’t know which 600 bits. Eve can see all communication.

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Extractors in Cryptography Alice and Bob compute a shared secret string of 300 bits, about which Eve has negligible information: To Eve, s appears like output of known bit- fixing source. So Ext(s) will appear almost random. Hence shared secret = Ext(s).

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