# Computational Notions of Entropy and Cryptographic Applications (US-Israel Binational Science Foundation; BSF 2010196)

2017
Steinke, Thomas, Salil Vadhan, and Andrew Wan. “Pseudorandomness and Fourier growth bounds for width 3 branching programs.” Theory of Computing – Special Issue on APPROX-RANDOM 2014 13, no. 12 (2017): 1-50. Publisher's VersionAbstract

Version History: a conference version of this paper appeared in the Proceedings of the 18th International Workshop on Randomization and Computation (RANDOM'14). Full version posted as ECCC TR14-076 and arXiv:1405.7028 [cs.CC].

We present an explicit pseudorandom generator for oblivious, read-once, width-3 branching programs, which can read their input bits in any order. The generator has seed length $$Õ(\log^3 n)$$.The previously best known seed length for this model is $$n^{1/2+o(1)}$$ due to Impagliazzo, Meka, and Zuckerman (FOCS ’12). Our work generalizes a recent result of Reingold, Steinke, and Vadhan (RANDOM ’13) for permutation branching programs. The main technical novelty underlying our generator is a new bound on the Fourier growth of width-3, oblivious, read-once branching programs. Specifically, we show that for any $$f : \{0, 1\}^n → \{0, 1\}$$ computed by such a branching program, and $$k ∈ [n]$$,

$$\displaystyle\sum_{s⊆[n]:|s|=k} \big| \hat{f}[s] \big | ≤n^2 ·(O(\log n))^k$$,

where $$\hat{f}[s] = \mathbb{E}_U [f[U] \cdot (-1)^{s \cdot U}]$$ is the standard Fourier transform over $$\mathbb{Z}^n_2$$. The base $$O(\log n)$$ of the Fourier growth is tight up to a factor of $$\log \log n$$.

2014
Gopalan, Parikshit, Salil Vadhan, and Yuan Zhou. “Locally testable codes and Cayley graphs.” In In Moni Naor, editor, Innovations in Theoretical Computer Science (ITCS ‘14), 81-92. ACM, 2014. Publisher's VersionAbstract

Version History: Full version posted as https://arxiv.org/pdf/1308.5158.pdf.

We give two new characterizations of ($$\mathbb{F}_2$$-linear) locally testable error-correcting codes in terms of Cayley graphs over $$\mathbb{F}^h_2$$ :

1. A locally testable code is equivalent to a Cayley graph over $$\mathbb{F}^h_2$$ whose set of generators is significantly larger than $$h$$ and has no short linear dependencies, but yields a shortest-path metric that embeds into $$\ell_1$$ with constant distortion. This extends and gives a converse to a result of Khot and Naor (2006), which showed that codes with large dual distance imply Cayley graphs that have no low-distortion embeddings into $$\ell_1$$.

2. A locally testable code is equivalent to a Cayley graph over $$\mathbb{F}^h_2$$ that has significantly more than $$h$$ eigenvalues near 1, which have no short linear dependencies among them and which “explain” all of the large eigenvalues. This extends and gives a converse to a recent construction of Barak et al. (2012), which showed that locally testable codes imply Cayley graphs that are small-set expanders but have many large eigenvalues.

2013
Vadhan, Salil, and Colin Jia Zheng. “A uniform min-max theorem with applications in cryptography.” In Ran Canetti and Juan Garay, editors, Advances in Cryptology—CRYPTO ‘13, Lecture Notes on Computer Science, 8042:93-110. Springer Verlag, Lecture Notes in Computer Science, 2013. Publisher's VersionAbstract

Version History:

Full version published in ECCC2013 and IACR ePrint 2013.

We present a new, more constructive proof of von Neumann’s Min-Max Theorem for two-player zero-sum game — specifically, an algorithm that builds a near-optimal mixed strategy for the second player from several best-responses of the second player to mixed strategies of the first player. The algorithm extends previous work of Freund and Schapire (Games and Economic Behavior ’99) with the advantage that the algorithm runs in poly$$(n)$$ time even when a pure strategy for the first player is a distribution chosen from a set of distributions over $$\{0,1\}^n$$. This extension enables a number of additional applications in cryptography and complexity theory, often yielding uniform security versions of results that were previously only proved for nonuniform security (due to use of the non-constructive Min-Max Theorem).

We describe several applications, including a more modular and improved uniform version of Impagliazzo’s Hardcore Theorem (FOCS ’95), showing impossibility of constructing succinct non-interactive arguments (SNARGs) via black-box reductions under uniform hardness assumptions (using techniques from Gentry and Wichs (STOC ’11) for the nonuniform setting), and efficiently simulating high entropy distributions within any sufficiently nice convex set (extending a result of Trevisan, Tulsiani and Vadhan (CCC ’09)).

Reingold, Omer, Thomas Steinke, and Salil Vadhan. “Pseudorandomness for regular branching programs via Fourier analysis.” In Sofya Raskhodnikova and José Rolim, editors, Proceedings of the 17th International Workshop on Randomization and Computation (RANDOM ‘13), Lecture Notes in Computer Science, 8096:655-670. Springer-Verlag, 2013. Publisher's VersionAbstract

Version HistoryFull version posted as ECCC TR13-086 and arXiv:1306.3004 [cs.CC].

We present an explicit pseudorandom generator for oblivious, read-once, permutation branching programs of constant width that can read their input bits in any order. The seed length is $$O(\log^2n)$$, where $$n$$ is the length of the branching program. The previous best seed length known for this model was $$n^{1/2+o(1)}$$, which follows as a special case of a generator due to Impagliazzo, Meka, and Zuckerman (FOCS 2012) (which gives a seed length of $$s^{1/2+o(1)}$$ for arbitrary branching programs of size $$s$$). Our techniques also give seed length $$n^{1/2+o(1)}$$ for general oblivious, read-once branching programs of width $$2^{n^{o(1)}}$$) , which is incomparable to the results of Impagliazzo et al.

Our pseudorandom generator is similar to the one used by Gopalan et al. (FOCS 2012) for read-once CNFs, but the analysis is quite different; ours is based on Fourier analysis of branching programs. In particular, we show that an oblivious, read-once, regular branching program of width $$w$$ has Fourier mass at most $$(2w^2)^k$$ at level $$k$$, independent of the length of the program.

2012
Vadhan, Salil, and Colin Jia Zheng. “Characterizing pseudoentropy and simplifying pseudorandom generator constructions.” In Proceedings of the 44th Annual ACM Symposium on Theory of Computing (STOC ‘12), 817-836. ACM, 2012. Publisher's VersionAbstract

Version HistoryFull version posted as ECCC TR11-141.

We provide a characterization of pseudoentropy in terms of hardness of sampling: Let ($$X, B$$) be jointly distributed random variables such that $$B$$ takes values in a polynomial-sized set. We show that $$B$$ is computationally indistinguishable from a random variable of higher Shannon entropy given $$X$$ if and only if there is no probabilistic polynomial-time $$S$$ such that $$(X, S(X))$$ has small KL divergence from $$(X, B)$$. This can be viewed as an analogue of the Impagliazzo Hard- core Theorem (FOCS ‘95) for Shannon entropy (rather than min-entropy).

Using this characterization, we show that if $$f$$ is a one-way function, then $$(f(U_n), U_n)$$ has “next-bit pseudoentropy” at least $$n + \log n$$, establishing a conjecture of Haitner, Reingold, and Vadhan (STOC ‘10). Plugging this into the construction of Haitner et al., this yields a simpler construction of pseudorandom generators from one-way functions. In particular, the construction only performs hashing once, and only needs the hash functions that are randomness extractors (e.g. universal hash functions) rather than needing them to support “local list-decoding” (as in the Goldreich–Levin hardcore predicate, STOC ‘89).

With an additional idea, we also show how to improve the seed length of the pseudorandom generator to $$\tilde{O}(n^3)$$, compared to $$\tilde{O}(n^4)$$ in the construction of Haitner et al.