**The Chaining Lemma and its application**

*Ivan Damgaard and Sebastian Faust and Pratyay Mukherjee and Daniele Venturi*

**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 (T1,....Tt) which are applied in succession to that source to generate the chain X[0]-->X[1]-->.....-->X[t] such that X[i] = Ti(X[i-1]). 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 the preceding part remains highly random. We believe 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.

**Category / Keywords: **foundations / information theory, tamper resistance

**Original Publication**** (with major differences): **ICITS 2015

**Date: **received 2 Dec 2014, last revised 18 Feb 2015

**Contact author: **pratyay85 at gmail com

**Available format(s): **PDF | BibTeX Citation

**Version: **20150218:103630 (All versions of this report)

**Short URL: **ia.cr/2014/979

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