One of the most studied algorithmic problems on lattices is the shortest vector problem SVP : given a lattice, find the shortest non-zero vector in it. In particular, we prove the NP-hardness of approximating SVP in the Euclidean norm 12 within any factor less than [square root of]2.
The same NP-hardness results hold for deterministic non-uniform reductions. A deterministic uniform reduction is also given under a reasonable number theoretic conjecture concerning the distribution of smooth numbers.
In proving the NP-hardness of SVP we develop a number of technical tools that might be of independent interest. Description Thesis Ph. Date issued Department Massachusetts Institute of Technology. Department of Electrical Engineering and Computer Science.
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One of the most studied algorithmic problems on lattices is the shortest vector problem SVP : given a lattice, find the shortest non-zero vector in it. In particular, we prove the NP-hardness of approximating SVP in the Euclidean norm 12 within any factor less than [square root of]2. The same NP-hardness results hold for deterministic non-uniform reductions. A deterministic uniform reduction is also given under a reasonable number theoretic conjecture concerning the distribution of smooth numbers.
In proving the NP-hardness of SVP we develop a number of technical tools that might be of independent interest. The common approach to analyzing a concrete cryptographic primitive is to analyze the performance of known algorithms to estimate the attack complexity of a hypothetical adversary. This requires a thorough theoretical understanding of the best performing algorithms.
Unfortunately, for many subclasses of lattice algorithms there is a gap in our understanding, which leads to problems in the cryptanalytic process. In this part of the thesis we address these issues in two closely related subclasses of such algorithms. We develop new algorithms and analyze existing ones and show that in both cases it is possible to obtain algorithms that are simultaneously well understood in theory and competitive in practice.
In Part II we focus on an integral part of most lattice-based schemes: sampling from a specific distribution over the integers. Implementing such a sampler securely and efficiently can be challenging for distributions commonly used in lattice-based schemes. We introduce new tools and security proofs that reduce the precision requirements for samplers, allowing more efficient implementations in a wide range of settings while maintaining high levels of security.
Finally, we propose a new sampling algorithms with a unique set of properties desirable for implementations of cryptographic primitives. Skip to main content. Email Facebook Twitter. Abstract Lattice-based cryptography is an extraordinarily popular subfield of cryptography. Thumbnails Document Outline Attachments. Highlight all Match case.
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Certified Writers Our writers hold Ph. Original Papers We have zero tolerance for plagiarism; thus we guarantee that every paper is written from scratch. Prompt Delivery All papers are delivered on time, even if your deadline is tight! These innovations led to the development of much more efficient somewhat and fully homomorphic cryptosystems. These include:. This NTRU variant was subsequently shown vulnerable to subfield lattice attacks,   which is why these two schemes are no longer used in practice.
All the second-generation cryptosystems still follow the basic blueprint of Gentry's original construction, namely they first construct a somewhat homomorphic cryptosystem and then convert it to a fully homomorphic cryptosystem using bootstrapping.
A distinguishing characteristic of the second-generation cryptosystems is that they all feature a much slower growth of the noise during the homomorphic computations. Another distinguishing feature of second-generation schemes is that they are efficient enough for many applications even without invoking bootstrapping, instead operating in the leveled FHE mode. FHEW introduced a new method to compute Boolean gates on encrypted data that greatly simplifies bootstrapping, and implemented a variant of the bootstrapping procedure.
CKKS scheme  supports efficient rounding operations in encrypted state. The rounding operation controls noise increase in encrypted multiplication, which reduces the number of bootstrapping in a circuit. This is due to a characteristic of CKKS scheme that encrypts approximate values rather than exact values.
When computers store real-valued data, they remember approximate values with long significant bits, not real values exactly. CKKS scheme is constructed to deal efficiently with the errors arising from the approximations. The scheme is familiar to machine learning which has inherent noises in its structure.
The authors also propose mitigation strategies for these attacks, and include a Responsible Disclosure in the paper suggesting that the homomorphic encryption libraries already implemented mitigations for the attacks before the article became publicly available. Further information on the mitigation strategies implemented in the homomorphic encryption libraries has also been published.
The homomorphic property is then. A cryptosystem that supports arbitrary computation on ciphertexts is known as fully homomorphic encryption FHE. Such a scheme enables the construction of programs for any desirable functionality, which can be run on encrypted inputs to produce an encryption of the result.
Since such a program need never decrypt its inputs, it can be run by an untrusted party without revealing its inputs and internal state. Fully homomorphic cryptosystems have great practical implications in the outsourcing of private computations, for instance, in the context of cloud computing.
An up-to-date list of homomorphic encryption implementations is also maintained by the community on GitHub. There are several open-source implementations of second- and third-generation fully homomorphic encryption schemes. Third-generation FHE scheme implementations often bootstrap after each Boolean gate operation but have limited support for packing and efficient arithmetic computations; they are typically used to compute Boolean circuits over encrypted bits.
The choice of using a second-generation vs. A community standard for homomorphic encryption is maintained by the HomomorphicEncryption. The current standard document includes specifications of secure parameters for RLWE. From Wikipedia, the free encyclopedia. Form of encryption that allows computation on ciphertexts. Private set intersection Functional encryption. Rivest, L. Adleman, and M. On data banks and privacy homomorphisms.
In Foundations of Secure Computation , ISBN S2CID Boneh, E. Goh, and K. In Theory of Cryptography Conference , Ishai and A. Evaluating branching programs on encrypted data. Eurocrypt Archived from the original on Crypto Lecture Notes in Computer Science. PKC Brakerski, C. Gentry, and V. Brakerski and V. Lopez-Alt, E.
Tromer, and V. Bos, K. Lauter, J. Loftus, and M. Albrecht, S. Bai, and L. Gentry, S. Halevi, and N. Fully Homomorphic Encryption with Polylog Overhead. Better Bootstrapping in Fully Homomorphic Encryption. Designs, Codes and Cryptography. Gentry, A. Sahai, and B. Alperin-Sheriff and C. Faster Bootstrapping with Polynomial Error. Retrieved 31 December Gama, M. Nguyen, and X.