Computing generator in cyclotomic integer rings

Computing Generator in Cyclotomic Integer Rings - A Subfield Algorithm for the Principal Ideal Problem in L|Δ𝕂|(½) and Application to the Cryptanalysis of a FHE Scheme

Computing generator in cyclotomic integer rings A L|∆K| (1/2) algorithm for the Principal Ideal Problem and application to the cryptanalysis of a FHE scheme

The Principal Ideal Problem (resp. Short Principal Ideal Problem), shorten as PIP (resp. SPIP), consists in finding a generator (resp. short generator) of a principal ideal in the ring of integers of a number field. Several lattice-based cryptosystems rely on the presumed hardness of these two problems. Yet, in practice, most of them do not use an arbitrary number field but a power-of-two cyclo...

Polynomial Time Reduction from Approximate Shortest Vector Problem to Principal Ideal Problem for Lattices in Some Cyclotomic Rings

Many cryptographic schemes have been established based on the hardness of lattice problems. For the asymptotic efficiency, ideal lattices in the ring of cyclotomic integers are suggested to be used in most such schemes. On the other hand in computational algebraic number theory one of the main problem is the principal ideal problem (PIP). Its goal is to find a generator of any principal ideal i...

Subring Homomorphic Encryption

In this paper, we construct subring homomorphic encryption scheme that is a homomorphic encryption scheme built on the decomposition ring, which is a subring of cyclotomic ring. In the scheme, each plaintext slot contains an integer in Zpl , rather than an element of GF(p) as in conventional homomorphic encryption schemes on cyclotomic rings. Our benchmark results indicate that the subring homo...

Sequences related to Legendre/Jacobi sequences

Two families of binary sequences are constructed from dth order cyclotomic generator and from dth order generalized cyclotomic generator with respect to two distinct primes respectively. By using estimates of certain exponential sums over rings or fields, the upper bounds of both the well-distribution measure and the order k (at least for small k) correlation measure of the resulting binary seq...