Small-bias sample space

In theoretical computer science, a small-bias sample space (also known as $\epsilon$-biased sample space, $\epsilon$-biased generator, or small-bias probability space) is a probability distribution that fools parity functions. In other words, no parity function can distinguish between a small-bias sample space and the uniform distribution with high probability, and hence, small-bias sample spaces naturally give rise to pseudorandom generators for parity functions.

The main useful property of small-bias sample spaces is that they need far fewer truly random bits than the uniform distribution to fool parities. Efficient constructions of small-bias sample spaces have found many applications in computer science, some of which are derandomization, error-correcting codes, and probabilistically checkable proofs. The connection with error-correcting codes is in fact very strong since $\epsilon$-biased sample spaces are equivalent to $\epsilon$-balanced error-correcting codes.

Definition

Bias

Let $X$ be a probability distribution over $\{0,1\}^{n}$. The bias of $X$ with respect to a set of indices $I\subseteq \{1,\dots ,n\}$ is defined as

${\text{bias}}_{I}(X)=\left|\Pr _{x\sim X}\left(\sum _{i\in I}x_{i}=0\right)-\Pr _{x\sim X}\left(\sum _{i\in I}x_{i}=1\right)\right|=\left|2\cdot \Pr _{x\sim X}\left(\sum _{i\in I}x_{i}=0\right)-1\right|\,,$

where the sum is taken over $\mathbb {F} _{2}$, the finite field with two elements. In other words, the sum $\sum _{i\in I}x_{i}$ equals $0$ if the number of ones in the sample $x\in \{0,1\}^{n}$ at the positions defined by $I$ is even, and otherwise, the sum equals $1$. For $I=\emptyset$, the empty sum is defined to be zero, and hence ${\text{bias}}_{\emptyset }(X)=1$.

ϵ-biased sample space

A probability distribution $X$ over $\{0,1\}^{n}$ is called an $\epsilon$-biased sample space if ${\text{bias}}_{I}(X)\leq \epsilon$ holds for all non-empty subsets $I\subseteq \{1,2,\ldots ,n\}$.

ϵ-biased set

An $\epsilon$-biased sample space $X$ that is generated by picking a uniform element from a multiset $X\subseteq \{0,1\}^{n}$ is called $\epsilon$-biased set. The size $s$ of an $\epsilon$-biased set $X$ is the size of the multiset that generates the sample space.

ϵ-biased generator

An $\epsilon$-biased generator $G:\{0,1\}^{\ell }\to \{0,1\}^{n}$ is a function that maps strings of length $\ell$ to strings of length $n$ such that the multiset $X_{G}=\{G(y)\;\vert \;y\in \{0,1\}^{\ell }\}$ is an $\epsilon$-biased set. The seed length of the generator is the number $\ell$ and is related to the size of the $\epsilon$-biased set $X_{G}$ via the equation $s=2^{\ell }$.

Connection with epsilon-balanced error-correcting codes

There is a close connection between $\epsilon$-biased sets and $\epsilon$-balanced linear error-correcting codes. A linear code $C:\{0,1\}^{n}\to \{0,1\}^{s}$ of message length $n$ and block length $s$ is $\epsilon$-balanced if the Hamming weight of every nonzero codeword $C(x)$ is between $({\frac {1}{2}}-\epsilon )s$ and $({\frac {1}{2}}+\epsilon )s$. Since $C$ is a linear code, its generator matrix is an $(n\times s)$-matrix $A$ over $\mathbb {F} _{2}$ with $C(x)=x\cdot A$.

Then it holds that a multiset $X\subset \{0,1\}^{n}$ is $\epsilon$-biased if and only if the linear code $C_{X}$, whose columns are exactly elements of $X$, is $\epsilon$-balanced.

Constructions of small epsilon-biased sets

Usually the goal is to find $\epsilon$-biased sets that have a small size $s$ relative to the parameters $n$ and $\epsilon$. This is because a smaller size $s$ means that the amount of randomness needed to pick a random element from the set is smaller, and so the set can be used to fool parities using few random bits.

Theoretical bounds

The probabilistic method gives a non-explicit construction that achieves size $s=O(n/\epsilon ^{2})$. The construction is non-explicit in the sense that finding the $\epsilon$-biased set requires a lot of true randomness, which does not help towards the goal of reducing the overall randomness. However, this non-explicit construction is useful because it shows that these efficient codes exist. On the other hand, the best known lower bound for the size of $\epsilon$-biased sets is $s=\Omega (n/(\epsilon ^{2}\log(1/\epsilon ))$, that is, in order for a set to be $\epsilon$-biased, it must be at least that big.

Explicit constructions

There are many explicit, i.e., deterministic constructions of $\epsilon$-biased sets with various parameter settings:

• Naor & Naor (1990) achieve $\displaystyle s={\frac {n}{{\text{poly}}(\epsilon )}}$. The construction makes use of Justesen codes (which is a concatenation of Reed–Solomon codes with the Wozencraft ensemble) as well as expander walk sampling.
• Alon et al. (1992) achieve $\displaystyle s=O\left({\frac {n}{\epsilon \log(n/\epsilon )}}\right)^{2}$. One of their constructions is the concatenation of Reed–Solomon codes with the Hadamard code; this concatenation turns out to be an $\epsilon$-balanced code, which gives rise to an $\epsilon$-biased sample space via the connection mentioned above.
• Concatenating Algebraic geometric codes with the Hadamard code gives an $\epsilon$-balanced code with $\displaystyle s=O\left({\frac {n}{\epsilon ^{3}\log(1/\epsilon )}}\right)$.
• Ben-Aroya & Ta-Shma (2009) achieves $\displaystyle s=O\left({\frac {n}{\epsilon ^{2}\log(1/\epsilon )}}\right)^{5/4}$.

These bounds are mutually incomparable. In particular, none of these constructions yields the smallest $\epsilon$-biased sets for all settings of $\epsilon$ and $n$.

Application: almost k-wise independence

An important application of small-bias sets lies in the construction of almost k-wise independent sample spaces.

k-wise independent spaces

A random variable $Y$ over $\{0,1\}^{n}$ is a k-wise independent space if, for all index sets $I\subseteq \{1,\dots ,n\}$ of size $k$, the marginal distribution $Y|_{I}$ is exactly equal to the uniform distribution over $\{0,1\}^{k}$. That is, for all such $I$ and all strings $z\in \{0,1\}^{k}$, the distribution $Y$ satisfies $\Pr _{Y}(Y|_{I}=z)=2^{-k}$.

Constructions and bounds

k-wise independent spaces are fairly well-understood.

• A simple construction by Joffe (1974) achieves size $n^{k}$.
• Alon, Babai & Itai (1986) construct a k-wise independent space whose size is $n^{k/2}$.
• Chor et al. (1985) prove that no k-wise independent space can be significantly smaller than $n^{k/2}$.

Joffe's construction

Joffe (1974) constructs a $k$-wise independent space $Y$ over the finite field with some prime number $n>k$ of elements, i.e., $Y$ is a distribution over $\mathbb {F} _{n}^{n}$. The initial $k$ marginals of the distribution are drawn independently and uniformly at random:

$(Y_{0},\dots ,Y_{k-1})\sim \mathbb {F} _{n}^{k}$.

For each $i$ with $k\leq i<n$, the marginal distribution of $Y_{i}$ is then defined as

$Y_{i}=Y_{0}+Y_{1}\cdot i+Y_{2}\cdot i^{2}+\dots +Y_{k-1}\cdot i^{k-1}\,,$

where the calculation is done in $\mathbb {F} _{n}$. Joffe (1974) proves that the distribution $Y$ constructed in this way is $k$-wise independent as a distribution over $\mathbb {F} _{n}^{n}$. The distribution $Y$ is uniform on its support, and hence, the support of $Y$ forms a $k$-wise independent set. It contains all $n^{k}$ strings in $\mathbb {F} _{n}^{k}$ that have been extended to strings of length $n$ using the deterministic rule above.

Almost k-wise independent spaces

A random variable $Y$ over $\{0,1\}^{n}$ is a $\delta$-almost k-wise independent space if, for all index sets $I\subseteq \{1,\dots ,n\}$ of size $k$, the restricted distribution $Y|_{I}$ and the uniform distribution $U_{k}$ on $\{0,1\}^{k}$ are $\delta$-close in 1-norm, i.e., ${\Big \|}Y|_{I}-U_{k}{\Big \|}_{1}\leq \delta$.

Constructions

Naor & Naor (1990) give a general framework for combining small k-wise independent spaces with small $\epsilon$-biased spaces to obtain $\delta$-almost k-wise independent spaces of even smaller size. In particular, let $G_{1}:\{0,1\}^{h}\to \{0,1\}^{n}$ be a linear mapping that generates a k-wise independent space and let $G_{2}:\{0,1\}^{\ell }\to \{0,1\}^{h}$ be a generator of an $\epsilon$-biased set over $\{0,1\}^{h}$. That is, when given a uniformly random input, the output of $G_{1}$ is a k-wise independent space, and the output of $G_{2}$ is $\epsilon$-biased. Then $G:\{0,1\}^{\ell }\to \{0,1\}^{n}$ with $G(x)=G_{1}(G_{2}(x))$ is a generator of an $\delta$-almost $k$-wise independent space, where $\delta =2^{k/2}\epsilon$.

As mentioned above, Alon, Babai & Itai (1986) construct a generator $G_{1}$ with $h={\tfrac {k}{2}}\log n$, and Naor & Naor (1990) construct a generator $G_{2}$ with $\ell =\log s=\log h+O(\log(\epsilon ^{-1}))$. Hence, the concatenation $G$ of $G_{1}$ and $G_{2}$ has seed length $\ell =\log k+\log \log n+O(\log(\epsilon ^{-1}))$. In order for $G$ to yield a $\delta$-almost k-wise independent space, we need to set $\epsilon =\delta 2^{-k/2}$, which leads to a seed length of $\ell =\log \log n+O(k+\log(\delta ^{-1}))$ and a sample space of total size $2^{\ell }\leq \log n\cdot {\text{poly}}(2^{k}\cdot \delta ^{-1})$.

References

• Alon, Noga; Babai, László; Itai, Alon (1986), "A fast and simple randomized parallel algorithm for the maximal independent set problem" (PDF), Journal of Algorithms, 7 (4): 567–583, doi:10.1016/0196-6774(86)90019-2 {{citation}}: Invalid |ref=harv (help)
• Alon, Noga; Goldreich, Oded; Håstad, Johan; Peralta, René (1992), "Simple Constructions of Almost k-wise Independent Random Variables" (PDF), Random Structures & Algorithms, 3 (3): 289–304, doi:10.1002/rsa.3240030308 {{citation}}: Invalid |ref=harv (help)
• Ben-Aroya, Avraham; Ta-Shma, Amnon (2009), "Constructing Small-Bias Sets from Algebraic-Geometric Codes" (PDF), Proceedings of the 50th Annual Symposium on Foundations of Computer Science, FOCS 2009: 191–197, doi:10.1109/FOCS.2009.44, ISBN 978-1-4244-5116-6 {{citation}}: Invalid |ref=harv (help)
• Chor, Benny; Goldreich, Oded; Håstad, Johan; Freidmann, Joel; Rudich, Steven; Smolensky, Roman (1985), "The bit extraction problem or t-resilient functions", Proceedings of the 26th Annual Symposium on Foundations of Computer Science, FOCS 1985: 396–407, doi:10.1109/SFCS.1985.55, ISBN 0-8186-0644-4 {{citation}}: Invalid |ref=harv (help)
• Goldreich, Oded (2001), Lecture 7: Small bias sample spaces {{citation}}: Invalid |ref=harv (help)
• Joffe, Anatole (1974), "On a Set of Almost Deterministic k-Independent Random Variables", Annals of Probability, 2 (1): 161–162, doi:10.1214/aop/1176996762 {{citation}}: Invalid |ref=harv (help)
• Naor, Joseph; Naor, Moni (1990), "Small-bias Probability Spaces: efficient constructions and Applications", Proceedings of the 22nd annual ACM symposium on Theory of computing, STOC 1990: 213–223, doi:10.1145/100216.100244, ISBN 0897913612 {{citation}}: Invalid |ref=harv (help)