Min-entropy

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Last modified: 2026-11-12 01:15:28 (PST)

The entropy of a random variable averages its surprisal. Other summaries of its probabilities answer other questions. This page adds the smallest of them, the min-entropy, to the collision entropy.

1 Min-entropy

Definition 1 (Min-entropy) The min-entropy of a discrete random variable \(X\) is the logarithm of the reciprocal of the largest probability that \(X\) takes any one value:

\[{\mathbb{H}_{\infty}\mathopen{}\left(X\right)\mathclose{}} = -\log \max_x {\mathbb{P}\mathopen{}\left(X = x\right)\mathclose{}}.\]

It is the limiting case of the Rényi entropy of order \(\alpha\) (Rényi 1961), which approaches it as \(\alpha \to \infty\).

Example 1 (Min-entropy of a three-valued random variable)

Let \(X\) take three values with probabilities \(1/2\), \(1/4\) and \(1/4\). The R chunk below finds the largest probability and the min-entropy, with base-2 logarithms.

Show code

p <- c(1 / 2, 1 / 4, 1 / 4)
max_prob <- max(p)
min_entropy <- -log2(max_prob)
c(max_prob = max_prob, min_entropy = min_entropy)
#>    max_prob min_entropy 
#>         0.5         1.0

The largest probability is \(0.5\), so the min-entropy is \({\mathbb{H}_{\infty}\mathopen{}\left(X\right)\mathclose{}} = 1\) bit.

Theorem 1 (The min-entropy is at most the collision entropy) Let \(X\) be a discrete random variable. Then the min-entropy is at most the collision entropy:

\[{\mathbb{H}_{\infty}\mathopen{}\left(X\right)\mathclose{}} \le {\mathbb{H}_{2}\mathopen{}\left(X\right)\mathclose{}}.\]

Proof

Proof. Let \(p_{\max} = \max_x {\mathbb{P}\mathopen{}\left(X = x\right)\mathclose{}}\). Each probability \({\mathbb{P}\mathopen{}\left(X = x\right)\mathclose{}}\) is at most \(p_{\max}\), so \({\mathbb{P}\mathopen{}\left(X = x\right)\mathclose{}}^2 \le p_{\max} \, {\mathbb{P}\mathopen{}\left(X = x\right)\mathclose{}}\). Sum over the values \(x\):

\[\sum_x {\mathbb{P}\mathopen{}\left(X = x\right)\mathclose{}}^2 \le p_{\max} \sum_x {\mathbb{P}\mathopen{}\left(X = x\right)\mathclose{}}.\]

The probabilities sum to \(1\), so the collision probability is at most \(p_{\max}\):

\[\mathbb{P}(X = X') = \sum_x {\mathbb{P}\mathopen{}\left(X = x\right)\mathclose{}}^2 \le p_{\max}.\]

The function \(-\log\) is decreasing, so taking it of both sides flips the comparison:

\[-\log \mathbb{P}(X = X') \ge -\log p_{\max}.\]

The left side is \({\mathbb{H}_{2}\mathopen{}\left(X\right)\mathclose{}}\) and the right side is \({\mathbb{H}_{\infty}\mathopen{}\left(X\right)\mathclose{}}\).

Example 2 (Three entropies of one random variable)

The R chunk below computes the min-entropy, the collision entropy and the entropy of the random variable of Example 1.

Show code

p <- c(1 / 2, 1 / 4, 1 / 4)
min_entropy <- -log2(max(p))
collision_entropy <- -log2(sum(p^2))
shannon_entropy <- -sum(p * log2(p))
c(
  min_entropy = min_entropy,
  collision_entropy = collision_entropy,
  shannon_entropy = shannon_entropy
)
#>       min_entropy collision_entropy   shannon_entropy 
#>          1.000000          1.415037          1.500000

The values rise from \(1\) to \(1.415\) to \(1.5\) bits, as Theorem 1 and the collision entropy bound say.

Remark 1 (Min-entropy measures the chance of a single guess). The best single guess at the value of \(X\) is its most probable value, and that guess is right with probability \(2^{-{\mathbb{H}_{\infty}\mathopen{}\left(X\right)\mathclose{}}}\). So a min-entropy of \(k\) bits says that no guess succeeds with probability above \(2^{-k}\). The entropy gives no such guarantee, because it can be large while one value is very likely. That is why min-entropy is the measure of a secret’s unpredictability.

References

Rényi, Alfréd. 1961. “On Measures of Entropy and Information.” In Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1: Contributions to the Theory of Statistics, edited by Jerzy Neyman. University of California Press. https://projecteuclid.org/euclid.bsmsp/1200512181.