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TCC 2013: Theory of Cryptography pp 143–161Cite as

When Homomorphism Becomes a Liability

  • Conference paper

Part of the Lecture Notes in Computer Science book series (LNSC,volume 7785)

Abstract

We show that an encryption scheme cannot have a simple decryption function and be homomorphic at the same time, even with added noise. Specifically, if a scheme can homomorphically evaluate the majority function, then its decryption cannot be weakly-learnable (in particular, linear), even if the probability of decryption error is high. (In contrast, without homomorphism, such schemes do exist and are presumed secure, e.g. based on LPN.)

An immediate corollary is that known schemes that are based on the hardness of decoding in the presence of low hamming-weight noise cannot be fully homomorphic. This applies to known schemes such as LPN-based symmetric or public key encryption.

Using these techniques, we show that the recent candidate fully homomorphic encryption, suggested by Bogdanov and Lee (ePrint ’11, henceforth BL), is insecure. In fact, we show two attacks on the BL scheme: One that uses homomorphism, and another that directly attacks a component of the scheme.

Keywords

  • Encryption Scheme
  • Homomorphic Encryption
  • Decryption Function
  • Decryption Error
  • Cryptology ePrint Archive

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Authors and Affiliations

  1. Stanford University, USA

    Zvika Brakerski

Authors
  1. Zvika Brakerski
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Editor information

Editors and Affiliations

  1. UCLA, 3731E Boelter Hall, 90095, Los Angeles, CA, USA

    Amit Sahai

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© 2013 International Association for Cryptologic Research

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Brakerski, Z. (2013). When Homomorphism Becomes a Liability. In: Sahai, A. (eds) Theory of Cryptography. TCC 2013. Lecture Notes in Computer Science, vol 7785. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-36594-2_9

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  • DOI: https://doi.org/10.1007/978-3-642-36594-2_9

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-642-36593-5

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