Pseudorandomness for Linear Length Branching Programs and Stack Machines

Bogdanov, Andrej; Papakonstantinou, Periklis A.; Wan, Andrew

doi:10.1007/978-3-642-32512-0_38

Cited by 9 publications

(16 citation statements)

References 17 publications

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“…Our results hinge on the fact that "mixing" is well understood for regular ordered branching programs [9,36,28,14,45] and for (non-regular) width-2 ordered branching programs [5]. Indeed, understanding mixing underpins most results for restricted models of branching programs.…”

Section: Further Directions 71 Larger Widthmentioning

confidence: 69%

“…The classic result of Nisan [33] gave a generator stretching O(log 2 n) uniformly random bits to n bits that are pseudorandom against ordered branching programs of polynomial width. 1 Despite intensive study, this is the best known seed length for ordered branching programs even of width 3, but a variety of results have shown improvements for restricted classes of ordered branching programs such as width-2 programs [39,5], and "regular" or "permutation" ordered branching programs (of constant width) [9,10,28,14,45]. 2 For width 3, hitting set generators (a relaxation of pseudorandom generators) have been constructed [42,18].…”

Section: Pseudorandom Generators For Space-bounded Computationmentioning

confidence: 99%

“…In the study of pseudorandomness, it is well known that small-bias generators 5 with bias ε/L (which can be sampled using a seed of length O(log(n · L/ε)) [31,2]) will ε-fool any function whose Fourier mass is at most L. Width-2, read-once, oblivious branching programs have Fourier mass at most O(n) [5,39] and are thus fooled by small-bias generators with bias ε/n. Unfortunately, such a bound does not hold even for very simple width-3 programs.…”

Section: Fourier Growth Of Branching Programsmentioning

confidence: 99%

“…Unfortunately, such a bound does not hold even for very simple width-3 programs. For example, the "mod 3 function," which indicates when the Hamming weight of its input is a multiple of 3, has Fourier mass exponential in n. 5 A small-bias generator with bias µ outputs a random variable X ∈ {0, 1} n such that E X (−1) s·X ≤ µ for every s ⊂ [n] with s = / 0.…”

Section: Fourier Growth Of Branching Programsmentioning

confidence: 99%

“…It is known that for length-n, width-2, read-once, oblivious branching programs B we have L(B) ≤ O(n) [5], so they are fooled by 1/poly(n)-biased generators (which have seed length O(log n)). But for a length-n, width-3, permutation, read-once, oblivious branching program B we can have L(B) = 2 Θ(n) .…”

Section: Small-bias Distributionsmentioning

confidence: 99%

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Steinke¹,

Vadhan²,

Wan³

2017

Theory of Comput.

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Section: Further Directions 71 Larger Widthmentioning

confidence: 69%

Section: Pseudorandom Generators For Space-bounded Computationmentioning

confidence: 99%

Section: Fourier Growth Of Branching Programsmentioning

confidence: 99%

Section: Fourier Growth Of Branching Programsmentioning

confidence: 99%

Section: Small-bias Distributionsmentioning

confidence: 99%

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Steinke¹,

Vadhan²,

Wan³

2017

Theory of Comput.

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Provable Time-Memory Trade-Offs: Symmetric Cryptography Against Memory-Bounded Adversaries

Tessaro

Thiruvengadam

2018

Theory of Cryptography

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Hitting-Sets for ROABP and Sum of Set-Multilinear Circuits

Agrawal¹,

Gurjar²,

Korwar³

et al. 2015

SIAM J. Comput.

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Abstract. We give a n O(log n) -time (n is the input size) blackbox polynomial identity testing algorithm for unknown-order read-once oblivious algebraic branching programs (ROABP). The best time-complexity known for this class was n O(log 2 n) due to Forbes-Saptharishi-Shpilka (STOC 2014), and that too only for multilinear ROABP. We get rid of their exponential dependence on the individual degree. With this, we match the time-complexity for the unknown order ROABP with the known order ROABP (due to Forbes-Shpilka (FOCS 2013)) and also with the depth-3 setmultilinear circuits (due to Agrawal-Saha-Saxena (STOC 2013)). Our proof is simpler and involves a new technique called basis isolation.The depth-3 model has recently gained much importance, as it has become a stepping-stone to understanding general arithmetic circuits. Its restriction to multilinearity has known exponential lower bounds but no nontrivial blackbox identity tests. In this paper, we take a step towards designing such hitting-sets. We give the first subexponential whitebox PIT for the sum of constantly many setmultilinear depth-3 circuits. To achieve this, we define notions of distance and base sets. Distance, for a multilinear depth-3 circuit (say, in n variables and k product gates), measures how far are the partitions from a mere refinement. The 1-distance strictly subsumes the set-multilinear model, while n-distance captures general multilinear depth-3. We design a hitting-set in time (nk) O(∆ log n) for ∆-distance. Further, we give an extension of our result to models where the distance is large (close to n) but it is small when restricted to certain base sets (of variables).We also explore a new model of read-once algebraic branching programs (ROABP) where the factor-matrices are invertible (called invertible-factor ROABP). We design a hitting-set in time poly(n w 2 ) for width-w invertible-factor ROABP. Further, we could do without the invertibility restriction when w = 2. Previously, the best result for width-2 ROABP was quasi-polynomial time (Forbes-Saptharishi-Shpilka, STOC 2014).

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Pseudorandomness for Linear Length Branching Programs and Stack Machines

Cited by 9 publications

References 17 publications

Untitled

Untitled

Provable Time-Memory Trade-Offs: Symmetric Cryptography Against Memory-Bounded Adversaries

Hitting-Sets for ROABP and Sum of Set-Multilinear Circuits

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