The Chaining Lemma and Its Application

Abstract

We present a new information-theoretic result which we call the Chaining Lemma. It considers a so-called “chain” of random variables, defined by a source distribution X ⁽⁰⁾ with high min-entropy and a number (say, t in total) of arbitrary functions (T ₁,...,T _t ) which are applied in succession to that source to generate the chain X ⁽⁰⁾ $\underrightarrow{T_1}$ X ⁽¹⁾ $\underrightarrow{T_2}$ X ⁽²⁾... $\underrightarrow{T_t}$ X ^(t) . Intuitively, the Chaining Lemma guarantees that, if the chain is not too long, then either (i) the entire chain is “highly random”, in that every variable has high min-entropy; or (ii) it is possible to find a point j (1 ≤ j ≤ t) in the chain such that, conditioned on the end of the chain i.e. X ^(j) $\underrightarrow{T_{j+1}}$ X ^(j + 1) ... $\underrightarrow{T_t}$ X ^(t), the preceding part X ⁽⁰⁾ $\underrightarrow{T_{1}}$ X ⁽¹⁾ ... $\underrightarrow{T_j}$ X ^(j) remains highly random. We think this is an interesting information-theoretic result which is intuitive but nevertheless requires rigorous case-analysis to prove.

We believe that the above lemma will find applications in cryptography. We give an example of this, namely we show an application of the lemma to protect essentially any cryptographic scheme against memory-tampering attacks. We allow several tampering requests, the tampering functions can be arbitrary, however, they must be chosen from a bounded size set of functions that is fixed a priori.