Ntr¹u-like Public Key Cryptosystems beyond Dedekind Domain Up to Alternative Algebra

In this paper, we show that the fundamental concepts behind the Ntr¹u cryptosystem can be extended to a broader algebra than Dedekind domains. Also, we present an abstract and generalized algorithm for constructing a Ntr¹u-like cryptosystem such that the underlying algebra can be non-commutative or even non-associative. To prove the main claim, we show that it is possible to generalize Ntr¹u over non-commutative Quaternions (algebra in the sense of Cayley-Dikson, of dimension four over an arbitrary principal ideal domain) as well as non-associative Octonions (a power-associative and alternative algebra of dimension eight over a principal ideal domain). Given the serious challenges ahead of non-commutative/non-associative algebra in quater- nionic or octonionic lattices, the proposed cryptosystems are more resistant to lattice-based attacks when compared to Ntr¹u. Concisely, this paper is making an abstract image of the mathematical base of Ntr¹u in such a way that one can make a similar cryptosystem based on various algebraic structures with the goal of better security against lattice attack and/or more capability for protocol design

A Non-commutative Cryptosystem Based on Quaternion Algebras

We propose BQTRU, a non-commutative NTRU-like cryptosystem over quaternion algebras. This cryptosystem uses bivariate polynomials as the underling ring. The multiplication operation in our cryptosystem can be performed with high speed using quaternions algebras over finite rings. As a consequence, the key generation and encryption process of our cryptosystem is faster than NTRU in comparable parameters. Typically using Strassen's method, the key generation and encryption process is approximately $16/7$ times faster than NTRU for an equivalent parameter set. Moreover, the BQTRU lattice has a hybrid structure that makes inefficient standard lattice attacks on the private key. This entails a higher computational complexity for attackers providing the opportunity of having smaller key sizes. Consequently, in this sense, BQTRU is more resistant than NTRU against known attacks at an equivalent parameter set. Moreover, message protection is feasible through larger polynomials and this allows us to obtain the same security level as other NTRU-like cryptosystems but using lower dimensions.Comment: Submitted for possible publicatio

Applying Fully Homomorphic Encryption: Practices and Problems

Fully homomorphic encryption (FHE) has been regarded as the "holy grail" of cryptography for its versatility as a cryptographic primitive and wide range of potential applications. Since Gentry published the first theoretically feasible FHE design in 2008, there has been a lot of new discoveries and inventions in this particular field. New schemes significantly reduce the computational cost of FHE and make practical deployment within reach. As a result, FHE schemes have come off the paper and been explored and tested extensively in practice. However, FHE is made possible with many new problems and assumptions that are not yet well studied. In this thesis we present a comprehensive and intuitive overview of the current applied FHE landscape, from design to implementation, and draw attention to potential vulnerabilities both in theory and in practice. In more detail, we show how to use currently available FHE libraries for aggregation and select parameters to avoid weak FHE instances

Towards practical fully homomorphic encryption

Fully homomorphic encryption (FHE) allows for computation of arbitrary func- tions on encrypted data by a third party, while keeping the contents of the encrypted data secure. This area of research has exploded in recent years following Gentry’s seminal work. However, the early realizations of FHE, while very interesting from a theoretical and proof-of-concept perspective, are unfortunately far too inefficient to provide any use in practice. The bootstrapping step is the main bottleneck in current FHE schemes. This step refreshes the noise level present in the ciphertexts by homomorphically evaluating the scheme’s decryption function over encryptions of the secret key. Bootstrapping is necessary in all known FHE schemes in order to allow an unlimited amount of computation, as without bootstrapping, the noise in the ciphertexts eventually grows to a point where decryption is no longer guaranteed to be correct. In this work, we present two new bootstrapping algorithms for FHE schemes. The first works on packed ciphertexts, which encrypt many bits at a time, while the second works on unpacked ciphertexts, which encrypt a single bit at a time. Our algorithms lie at the heart of the fastest currently existing implementations of fully homomorphic encryption for packed ciphertexts and for single-bit encryptions, respectively, running hundreds of times as fast for practical parameters as the previous best implementations.Ph.D

International Congress of Mathematicians: 2022 July 6–14: Proceedings of the ICM 2022

Following the long and illustrious tradition of the International Congress of Mathematicians, these proceedings include contributions based on the invited talks that were presented at the Congress in 2022. Published with the support of the International Mathematical Union and edited by Dmitry Beliaev and Stanislav Smirnov, these seven volumes present the most important developments in all fields of mathematics and its applications in the past four years. In particular, they include laudations and presentations of the 2022 Fields Medal winners and of the other prestigious prizes awarded at the Congress. The proceedings of the International Congress of Mathematicians provide an authoritative documentation of contemporary research in all branches of mathematics, and are an indispensable part of every mathematical library