Randomness extractor

The min-entropy of a distribution ${\displaystyle X}$ (denoted ${\displaystyle H_{\infty }(X)}$), is the largest real number ${\displaystyle k}$ such that ${\displaystyle \Pr[X=x]\leq 2^{-k}}$ for every ${\displaystyle x}$ in the range of ${\displaystyle X}$. In essence, this measures how likely ${\displaystyle X}$ is to take its most likely value, giving a worst-case bound on how random ${\displaystyle X}$ appears. Letting ${\displaystyle U_{\ell }}$ denote the uniform distribution over ${\displaystyle \{0,1\}^{\ell }}$, clearly ${\displaystyle H_{\infty }(U_{\ell })=\ell }$.

For an n-bit distribution ${\displaystyle X}$ with min-entropy k, we say that ${\displaystyle X}$ is an ${\displaystyle (n,k)}$ distribution.

Definition (Extractor): (k, ε)-extractor

Let ${\displaystyle {\text{Ext}}:\{0,1\}^{n}\times \{0,1\}^{d}\to \{0,1\}^{m}}$ be a function that takes as input a sample from an ${\displaystyle (n,k)}$ distribution ${\displaystyle X}$ and a d-bit seed from ${\displaystyle U_{d}}$, and outputs an m-bit string. ${\displaystyle {\text{Ext}}}$ is a (k, ε)-extractor, if for all ${\displaystyle (n,k)}$ distributions ${\displaystyle X}$, the output distribution of ${\displaystyle {\text{Ext}}}$ is ε-close to ${\displaystyle U_{m}}$.

In the above definition, ε-close refers to statistical distance.

Intuitively, an extractor takes a weakly random n-bit input and a short, uniformly random seed and produces an m-bit output that looks uniformly random. The aim is to have a low ${\displaystyle d}$ (i.e. to use as little uniform randomness as possible) and as high an ${\displaystyle m}$ as possible (i.e. to get out as many close-to-random bits of output as we can).

One of the most important aspects of cryptography is random key generation.[6] It is often necessary to generate secret and random keys from sources that are semi-secret or which may be compromised to some degree. By taking a single, short (and secret) random key as a source, an extractor can be used to generate a longer pseudo-random key, which then can be used for public key encryption. More specifically, when a strong extractor is used its output will appear be uniformly random, even to someone who sees part (but not all) of the source. For example, if the source is known but the seed is not known (or vice versa). This property of extractors is particularly useful in what is commonly called Exposure-Resilient cryptography in which the desired extractor is used as an Exposure-Resilient Function (ERF). Exposure-Resilient cryptography takes into account that the fact that it is difficult to keep secret the initial exchange of data which often takes place during the initialization of an encryption application e.g., the sender of encrypted information has to provide the receivers with information which is required for decryption.

The following paragraphs define and establish an important relationship between two kinds of ERF--k-ERF and k-APRF--which are useful in Exposure-Resilient cryptography.

Definition (k-ERF): An adaptive k-ERF is a function ${\displaystyle f}$ where, for a random input ${\displaystyle r}$ , when a computationally unbounded adversary ${\displaystyle A}$ can adaptively read all of ${\displaystyle r}$ except for ${\displaystyle k}$ bits, ${\displaystyle |\Pr\{A^{r}(f(r))=1\}-\Pr\{A^{r}(R)=1\}|\leq \epsilon (n)}$ for some negligible function ${\displaystyle \epsilon (n)}$ (defined below).

The goal is to construct an adaptive ERF whose output is highly random and uniformly distributed. But a stronger condition is often needed in which every output occurs with almost uniform probability. For this purpose Almost-Perfect Resilient Functions (APRF) are used. The definition of an APRF is as follows:

Definition (k-APRF): A ${\displaystyle k=k(n)}$ APRF is a function ${\displaystyle f}$ where, for any setting of ${\displaystyle n-k}$ bits of the input ${\displaystyle r}$ to any fixed values, the probability vector ${\displaystyle p}$ of the output ${\displaystyle f(r)}$ over the random choices for the ${\displaystyle k}$ remaining bits satisfies ${\displaystyle |p_{i}-2^{-m}|<2^{-m}\epsilon (n)}$ for all ${\displaystyle i}$ and for some negligible function ${\displaystyle \epsilon (n)}$.

Kamp and Zuckerman[7] have proved a theorem stating that if a function ${\displaystyle f}$ is a k-APRF, then ${\displaystyle f}$ is also a k-ERF. More specifically, any extractor having sufficiently small error and taking as input an oblivious, bit-fixing source is also an APRF and therefore also a k-ERF. A more specific extractor is expressed in this lemma:

Lemma: Any ${\displaystyle 2^{-m}\epsilon (n)}$-extractor ${\displaystyle f:\{0,1\}^{n}\rightarrow \{0,1\}^{m}}$ for the set of ${\displaystyle (n,k)}$ oblivious bit-fixing sources, where ${\displaystyle \epsilon (n)}$ is negligible, is also a k-APRF.

This lemma is proved by Kamp and Zuckerman.[7] The lemma is proved by examining the distance from uniform of the output, which in a ${\displaystyle 2^{-m}\epsilon (n)}$-extractor obviously is at most ${\displaystyle 2^{-m}\epsilon (n)}$, which satisfies the condition of the APRF.

The lemma leads to the following theorem, stating that there in fact exists a k-APRF function as described:

Theorem (existence): For any positive constant ${\displaystyle \gamma \leq {\frac {1}{2}}}$, there exists an explicit k-APRF ${\displaystyle f:\{0,1\}^{n}\rightarrow \{0,1\}^{m}}$, computable in a linear number of arithmetic operations on ${\displaystyle m}$-bit strings, with ${\displaystyle m=\Omega (n^{2\gamma })}$ and ${\displaystyle k=n^{{\frac {1}{2}}+\gamma }}$.

Definition (negligible function): In the proof of this theorem, we need a definition of a negligible function. A function ${\displaystyle \epsilon (n)}$ is defined as being negligible if ${\displaystyle \epsilon (n)=O\left({\frac {1}{n^{c}}}\right)}$ for all constants ${\displaystyle c}$.

Proof: Consider the following ${\displaystyle \epsilon }$-extractor: The function ${\displaystyle f}$ is an extractor for the set of ${\displaystyle (n,\delta n)}$ oblivious bit-fixing source: ${\displaystyle f:\{0,1\}^{n}\rightarrow \{0,1\}^{m}}$. ${\displaystyle f}$ has ${\displaystyle m=\Omega (\delta ^{2}n)}$, ${\displaystyle \epsilon =2^{-cm}}$ and ${\displaystyle c>1}$.

The proof of this extractor's existence with ${\displaystyle \delta \leq 1}$, as well as the fact that it is computable in linear computing time on the length of ${\displaystyle m}$ can be found in the paper by Jesse Kamp and David Zuckerman (p. 1240).

That this extractor fulfills the criteria of the lemma is trivially true as ${\displaystyle \epsilon =2^{-cm}}$ is a negligible function.

The size of ${\displaystyle m}$ is:

${\displaystyle m=\Omega (\delta ^{2}n)=\Omega (n)\geq \Omega (n^{2\gamma })}$

Since we know ${\displaystyle \delta \leq 1}$ then the lower bound on ${\displaystyle m}$ is dominated by ${\displaystyle n}$. In the last step we use the fact that ${\displaystyle \gamma \leq {\frac {1}{2}}}$ which means that the power of ${\displaystyle n}$ is at most ${\displaystyle 1}$. And since ${\displaystyle n}$ is a positive integer we know that ${\displaystyle n^{2\gamma }}$ is at most ${\displaystyle n}$.

The value of ${\displaystyle k}$ is calculated by using the definition of the extractor, where we know:

${\displaystyle (n,k)=(n,\delta n)\Rightarrow k=\delta n}$

and by using the value of ${\displaystyle m}$ we have:

${\displaystyle m=\delta ^{2}n=n^{2\gamma }}$

Using this value of ${\displaystyle m}$ we account for the worst case, where ${\displaystyle k}$ is on its lower bound. Now by algebraic calculations we get:

${\displaystyle \delta ^{2}n=n^{2\gamma }}$

${\displaystyle \Rightarrow \delta ^{2}=n^{2\gamma -1}}$

${\displaystyle \Rightarrow \delta =n^{\gamma -{\frac {1}{2}}}}$

Which inserted in the value of ${\displaystyle k}$ gives

${\displaystyle k=\delta n=n^{\gamma -{\frac {1}{2}}}n=n^{\gamma +{\frac {1}{2}}}}$,

which proves that there exists an explicit k-APRF extractor with the given properties. ${\displaystyle \Box }$

Randomness extractors are used widely in cryptographic applications, whereby a cryptographic hash function is applied to a high-entropy, but non-uniform source, such as disk drive timing information or keyboard delays, to yield a uniformly random result.

Randomness extractors have played a part in recent developments in quantum cryptography, where photons are used by the randomness extractor to generate secure random bits.

Randomness extraction is also used in some branches of computational complexity theory.

Random extraction is also used to convert data to a simple random sample, which is normally distributed, and independent, which is desired by statistics.

• Decorrelation
• Hardware random number generator
• Randomness merger
• Fuzzy extractor

• Randomness Extractors for Independent Sources and Applications, Anup Rao
• Recent developments in explicit constructions of extractors, Ronen Shaltiel
• Randomness Extraction and Key Derivation Using the CBC, Cascade and HMAC Modes, Yevgeniy Dodis et al.
• Key Derivation and Randomness Extraction, Olivier Chevassut et al.
• Deterministic Extractors for Bit-Fixing Sources and Exposure-Resilient Cryptography, Jesse Kamp and David Zuckerman
• Tossing a Biased Coin (and the optimality of advanced multi-level strategy) (lecture notes), Michael Mitzenmacher