Attacks on the Search-RLWE problem with small errors

The Ring Learning-With-Errors (RLWE) problem shows great promise for post-quantum cryptography and homomorphic encryption. We describe a new attack on the non-dual search RLWE problem with small error widths, using ring homomorphisms to finite fields and the chi-squared statistical test. In particular, we identify a "subfield vulnerability" (Section 5.2) and give a new attack which finds this vulnerability by mapping to a finite field extension and detecting non-uniformity with respect to the number of elements in the subfield. We use this attack to give examples of vulnerable RLWE instances in Galois number fields. We also extend the well-known search-to-decision reduction result to Galois fields with any unramified prime modulus q, regardless of the residue degree f of q, and we use this in our attacks. The time complexity of our attack is O(nq2f), where n is the degree of K and f is the residue degree of q in K. We also show an attack on the non-dual (resp. dual) RLWE problem with narrow error distributions in prime cyclotomic rings when the modulus is a ramified prime (resp. any integer). We demonstrate the attacks in practice by finding many vulnerable instances and successfully attacking them. We include the code for all attacks

On representations of dialgebras and conformal algebras

In this note, we observe a relation between dialgebras (in particular, Leibniz algebras) and conformal algebras. The purpose is to show how the methods of conformal algebras help solving problems on dialgebras, and, conversely, how the ideas of dialgebras work for conformal algebras.Comment: 11 page

Associated primes of graded components of local cohomology modules

The i-th local cohomology module of a finitely generated graded module M over a standard positively graded commutative Noetherian ring R with respect to the irrelevant ideal R+, is itself graded; all its graded components are finitely generated modules over R-0, the component of R of degree 0. It is known that the n-th component H-R+(i) (M)(n) of this local cohomology module H-R+(i) (M) is zero for all nmuch greater than0. This paper is concerned with the asymptotic behaviour of Ass(R0)(H-R+(i) (M)(n)) as n--> -infinity. The smallest i for which such study is interesting is the finiteness dimension f of M relative to R+, defined as the least integer j for which H-R+(j) (M) is not finitely generated. Brodmann and Hellus have shown that AssR(0)(H-R+(f) (M)(n)) is constant for all nmuch less than0 ( that is in their terminology AssR(0)(H-R+(f) (M)(n)) is asymptotically stable for n--> -infinity). The first main aim of this paper is to identify the ultimate constant value ( under the mild assumption that R is a homomorphic image of a regular ring) : our answer is precisely the set of contractions to R-0 of certain relevant primes of R whose existence is confirmed by Grothendieck's Finiteness Theorem for local cohomology. Brodmann and Hellus raised various questions about such asymptotic behaviour when i>f. They noted that Singh's study of a particular example ( in which f=2) shows that AssR(0)(H-R+(3) (R)(n)) need not be asymptotically stable for n--> -infinity. The second main aim of this paper is to determine, for Singh's example, AssR(0)(H-R+(3) (R)(n)) quite precisely for every integer n and, thereby answer one of the questions raised by Brodmann and Hellus

Approximation-based homomorphic encryption for secure and efficient blockchain-driven watermarking service

Homomorphic encryption has been widely used to preserve the privacy of watermarking process on blockchain-driven watermarking services. It offers transparent and traceable encrypted watermarking without revealing sensitive data such as original images or watermark data to the public. Nevertheless, the existing works suffer from enormous memory storage and extensive computing power. This study proposed an approximation-based homomorphic encryption for resource-efficient encrypted watermarking without sacrificing watermarking quality. We demonstrated the efficiency of the Cheon-Kim-Kim-Son (CKKS) encrypted watermarking process using discrete cosine transform-singular value decomposition (DCT-SVD) embedding. The evaluation results showed that it could preserve the watermarking quality similar to non-encrypted watermark embedding, even after geometrical and filtering attacks. Compared to existing homomorphic encryption, such as Brakerski-Gentry-Vaikuntanathan (BFV) encryption, it has superior performance regarding resource utilization and watermarking quality preservation