learning with errors

8 Submissions to publication!!! Code Offset in the Exponent

This paper is finally published.  Its my new record for number of needed submissions.  The worst part about it is that it was Luke’s first paper and this unnecessarily stunted his growth.


I think its very cool but I say that about everything I end up writing up!

Abstract: Fuzzy extractors transform a noisy source e into a stable key which can be reproduced from a nearby value e’. They are a fundamental tool for key derivation from biometric sources. This work introduces code offset in the exponent and uses this construction to build the first reusable fuzzy extractor that simultaneously supports structured, low entropy distributions with correlated symbols and confidence information. These properties are specifically motivated by the most pertinent applications—key derivation from biometrics and physical unclonable functions—which typically demonstrate low entropy with additional statistical correlations and benefit from extractors that can leverage confidence information for efficiency. Code offset in the exponent is a group encoding of the code offset construction (Juels and Wattenberg, CCS 1999) that stores the value e in a one-time pad which is sampled as a codeword, Ax, of a linear error-correcting code: Ax+ e. Rather than encoding Ax+ e directly, code offset in the exponent calls for encoding by exponentiation of a generator in a cryptographically strong group. We demonstrate security of the construction in the generic group model, establishing security whenever the inner product between the error distribution and all vectors in the null space of the code is unpredictable. We show this condition includes distributions supported by multiple prior fuzzy extractors. Our analysis also shows a prior construction of pattern matching obfuscation (Bishop et al., Crypto 2018) is secure for more distributions than previously known.

Computational Fuzzy Extractors

Benjamin Fuller, Xianrui Meng, and Leonid Reyzin.  Computational Fuzzy Extractors.  Asiacrypt 2013.


Fuzzy extractors derive strong keys from noisy sources. Their security is defined information- theoretically, which limits the length of the derived key, sometimes making it too short to be useful. We ask whether it is possible to obtain longer keys by considering computational security, and show the following.

-Negative Result: Noise tolerance in fuzzy extractors is usually achieved using an information reconciliation component called a “secure sketch.” The security of this component, which directly affects the length of the resulting key, is subject to lower bounds from coding theory. We show that, even when defined computationally, secure sketches are still subject to lower bounds from coding theory. Specifically, we consider two computational relaxations of the information-theoretic security requirement of secure sketches, using conditional HILL entropy and unpredictability entropy. For both cases we show that computational secure sketches cannot outperform the best information-theoretic secure sketches in the case of high-entropy Hamming metric sources.

-Positive Result: We show that the negative result can be overcome by analyzing computational fuzzy extractors directly. Namely, we show how to build a computational fuzzy extractor whose output key length equals the entropy of the source (this is impossible in the information-theoretic setting). Our construction is based on the hardness of the Learning with Errors (LWE) problem, and is secure when the noisy source is uniform or symbol-fixing (that is, each dimension is either uniform or fixed). As part of the security proof, we show a result of independent interest, namely that the decision version of LWE is secure even when a small number of dimensions has no error.