## Abstract

Several well-known public key encryption schemes, including those of Alekhnovich (FOCS 2003), Regev (STOC 2005), and Gentry, Peikert and Vaikuntanathan (STOC 2008), rely on the conjectured intractability of inverting noisy linear encodings. These schemes are limited in that they either require the underlying field to grow with the security parameter, or alternatively they can work over the binary field but have a low noise entropy that gives rise to sub-exponential attacks. Motivated by the goal of efficient public key cryptography, we study the possibility of obtaining improved security over the binary field by using different noise distributions. Inspired by an abstract encryption scheme of Micciancio (PKC 2010), we study an abstract encryption scheme that unifies all the three schemes mentioned above and allows for arbitrary choices of the underlying field and noise distributions. Our main result establishes an unexpected connection between the power of such encryption schemes and additive combinatorics. Concretely, we show that under the “approximate duality conjecture” from additive combinatorics (Ben-Sasson and Zewi, STOC 2011), every instance of the abstract encryption scheme over the binary field can be attacked in time (Formula presented.), where n is the maximum of the ciphertext size and the public key size (and where the latter excludes public randomness used for specifying the code). On the flip side, counter examples to the above conjecture (if false) may lead to candidate public key encryption schemes with improved security guarantees. We also show, using a simple argument that relies on agnostic learning of parities (Kalai, Mansour and Verbin, STOC 2008), that any such encryption scheme can be unconditionally attacked in time (Formula presented.), where n is the ciphertext size. Combining this attack with the security proof of Regev’s cryptosystem, we immediately obtain an algorithm that solves the learning parity with noise (LPN) problem in time (Formula presented.) using only n^{1+ɛ} samples, reproducing the result of Lyubashevsky (Random 2005) in a conceptually different way. Finally, we study the possibility of instantiating the abstract encryption scheme over constant-size rings to yield encryption schemes with no decryption error. We show that over the binary field decryption errors are inherent. On the positive side, building on the construction of matching vector families (Grolmusz, Combinatorica 2000; Efremenko, STOC 2009; Dvir, Gopalan and Yekhanin, FOCS 2010), we suggest plausible candidates for secure instances of the framework over constant-size rings that can offer perfectly correct decryption.

Original language | English |
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Title of host publication | Public-Key Cryptography – PKC 2016 - 19th IACR International Conference on Practice and Theory in Public-Key Cryptography, Proceedings |

Editors | Chen-Mou Cheng, Kai-Min Chung, Bo-Yin Yang, Giuseppe Persiano |

Publisher | Springer Verlag |

Pages | 417-446 |

Number of pages | 30 |

ISBN (Print) | 9783662493861 |

DOIs | |

State | Published - 2016 |

Externally published | Yes |

Event | 19th IACR International Conference on Practice and Theory in Public-Key Cryptography, PKC 2016 - Taipei, Taiwan, Province of China Duration: 6 Mar 2016 → 9 Mar 2016 |

### Publication series

Name | Lecture Notes in Computer Science (including subseries Lecture Notes in Artificial Intelligence and Lecture Notes in Bioinformatics) |
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Volume | 9615 |

ISSN (Print) | 0302-9743 |

ISSN (Electronic) | 1611-3349 |

### Conference

Conference | 19th IACR International Conference on Practice and Theory in Public-Key Cryptography, PKC 2016 |
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Country/Territory | Taiwan, Province of China |

City | Taipei |

Period | 6/03/16 → 9/03/16 |

### Bibliographical note

Funding Information:The research of the first two authors was supported by ERC grant no. 240258 (PaC) and ISF grant 1501/14.The research of the third author was supported by the CFEM center funded by the Danish Council for Strategic Research, the FP7 EU-project PRACTICE, the MPCPRO project funded by ERC and the CTIC center funded by the Danish National Research Foundation. The research of the fourth author was supported by ERC grant no. 259426 CaC, ISF grant 1709/14, and BSF grant 2012378. His research is also supported from a DARPA/ARL SAFEWARE award, NSF Frontier Award 1413955, NSF grants 1228984, 1136174, 1118096, and 1065276. This material is based upon work supported by the Defense Advanced Research Projects Agency through the ARL under Contract W911NF-15-C-0205. The views expressed are those of the author and do not reflect the official policy or position of the Department of Defense, the National Science Foundation, or the U.S. Government. The research of fifth author was partially supported by NSF grants CCF-1412958 and CCF-1445755 and the Rothschild fellowship.

Publisher Copyright:

© International Association for Cryptologic Research 2016.

## Keywords

- Additive combinatorics
- Learning parity with noise
- Noisy codewords
- Public key encryption

## ASJC Scopus subject areas

- Theoretical Computer Science
- Computer Science (all)