Three\u27s Compromised Too: Circular Insecurity for Any Cycle Length from (Ring-)LWE

Informally, a public-key encryption scheme is \emph{$k$-circular secure} if a cycle of~$k$ encrypted secret keys (\pkcenc_{\pk_{1}}(\sk_{2}), \pkcenc_{\pk_{2}}(\sk_{3}), \ldots, \pkcenc_{\pk_{k}}(\sk_{1})) is indistinguishable from encryptions of zeros. Circular security has applications in a wide variety of settings, ranging from security of symbolic protocols to fully homomorphic encryption. A fundamental question is whether standard security notions like IND-CPA/CCA imply $k$-circular security. For the case $k=2$, several works over the past years have constructed counterexamples---i.e., schemes that are CPA or even CCA secure but not $2$-circular secure---under a variety of well-studied assumptions (SXDH, decision linear, and LWE). However, for $k > 2$ the only known counterexamples are based on strong general-purpose obfuscation assumptions. In this work we construct $k$-circular security counterexamples for any $k \geq 2$ based on (ring-)LWE. Specifically: \begin{itemize} \item for any constant $k=O(1)$, we construct a counterexample based on $n$-dimensional (plain) LWE for \poly(n) approximation factors; \item for any k=\poly(\lambda), we construct one based on degree-$n$ ring-LWE for at most subexponential $\exp(n^{\varepsilon})$ factors. \end{itemize} Moreover, both schemes are $k\u27$-circular insecure for $2 \leq k\u27 \leq k$. Notably, our ring-LWE construction does not immediately translate to an LWE-based one, because matrix multiplication is not commutative. To overcome this, we introduce a new ``tensored\u27\u27 variant of LWE which provides the desired commutativity, and which we prove is actually equivalent to plain LWE

Learning with Errors is easy with quantum samples

Learning with Errors is one of the fundamental problems in computational learning theory and has in the last years become the cornerstone of post-quantum cryptography. In this work, we study the quantum sample complexity of Learning with Errors and show that there exists an efficient quantum learning algorithm (with polynomial sample and time complexity) for the Learning with Errors problem where the error distribution is the one used in cryptography. While our quantum learning algorithm does not break the LWE-based encryption schemes proposed in the cryptography literature, it does have some interesting implications for cryptography: first, when building an LWE-based scheme, one needs to be careful about the access to the public-key generation algorithm that is given to the adversary; second, our algorithm shows a possible way for attacking LWE-based encryption by using classical samples to approximate the quantum sample state, since then using our quantum learning algorithm would solve LWE

A Survey on Homomorphic Encryption Schemes: Theory and Implementation

Legacy encryption systems depend on sharing a key (public or private) among the peers involved in exchanging an encrypted message. However, this approach poses privacy concerns. Especially with popular cloud services, the control over the privacy of the sensitive data is lost. Even when the keys are not shared, the encrypted material is shared with a third party that does not necessarily need to access the content. Moreover, untrusted servers, providers, and cloud operators can keep identifying elements of users long after users end the relationship with the services. Indeed, Homomorphic Encryption (HE), a special kind of encryption scheme, can address these concerns as it allows any third party to operate on the encrypted data without decrypting it in advance. Although this extremely useful feature of the HE scheme has been known for over 30 years, the first plausible and achievable Fully Homomorphic Encryption (FHE) scheme, which allows any computable function to perform on the encrypted data, was introduced by Craig Gentry in 2009. Even though this was a major achievement, different implementations so far demonstrated that FHE still needs to be improved significantly to be practical on every platform. First, we present the basics of HE and the details of the well-known Partially Homomorphic Encryption (PHE) and Somewhat Homomorphic Encryption (SWHE), which are important pillars of achieving FHE. Then, the main FHE families, which have become the base for the other follow-up FHE schemes are presented. Furthermore, the implementations and recent improvements in Gentry-type FHE schemes are also surveyed. Finally, further research directions are discussed. This survey is intended to give a clear knowledge and foundation to researchers and practitioners interested in knowing, applying, as well as extending the state of the art HE, PHE, SWHE, and FHE systems.Comment: - Updated. (October 6, 2017) - This paper is an early draft of the survey that is being submitted to ACM CSUR and has been uploaded to arXiv for feedback from stakeholder