A Counterexample to the Chain Rule for Conditional HILL Entropy

A chain rule for an entropy notion

H

(·) states that the entropy

H

(

X

) of a variable

X

decreases by at most ℓ if conditioned on an ℓ-bit string

A

, i.e.,

H

(

X

|

A

) ≥ 

H

(

X

) − ℓ. More generally, it satisfies a chain rule for

conditional

entropy if

H

(

X

|

Y

,

A

) ≥ 

H

(

X

|

Y

) − ℓ.

All natural information theoretic entropy notions we are aware of (like Shannon or min-entropy) satisfy some kind of chain rule for conditional entropy. Moreover, many

computational

entropy notions (like Yao entropy, unpredictability entropy and several variants of HILL entropy) satisfy the chain rule for conditional entropy, though here not only the

quantity

decreases by ℓ, but also the

quality

of the entropy decreases exponentially in ℓ. However, for the standard notion of conditional HILL entropy (the computational equivalent of min-entropy) the existence of such a rule was unknown so far.

In this paper, we prove that for conditional HILL entropy no meaningful chain rule exists, assuming the existence of one-way permutations: there exist distributions

X

,

Y

,

A

, where

A

is a distribution over a

single

bit, but

H

HILL

(X|Y)≫

H

HILL

(X|Y,A), even if we simultaneously allow for a massive degradation in the quality of the entropy.

The idea underlying our construction is based on a surprising connection between the chain rule for HILL entropy and deniable encryption.