Modular multiplication is used in a wide range of applications. Most of the existing modular multiplication algorithms in the literature often focus on large size moduli. However, those large moduli oriented modular multiplication solutions are also used to implement modular arithmetic for applications requiring modular arithmetic on moduli of size inferior to a word size i.e. 32/64bits. As it happens, a large majority of applications are using word size modular arithmetic. In this work, we propose a new modular multiplication designed to be computed on one word size only. For word size moduli, in a large majority of instances, our solution outperforms other existing solutions including generalist solutions like Montgomery's and Barrett's modular multiplication as well as classes of moduli like Mersenne, Pseudo-Mersenne, Montgomery-Friendly and Generalized Mersenne.

The selection of the elements of the bases in an RNS modular multiplication method is crucial and has a great impact in the overall performance.This work proposes specific sets of optimal RNS moduli with elements of Hamming weight three whose inverses used in the MRS reconstruction have very small Hamming weight. This property is exploited in RNS bases conversions, to completely remove and replace the products by few additions/subtractions and shifts, reducing the time complexity of modular multiplication.These bases are specially crafted to computation with operands of sizes 256 or more and are suitable for cryptographic applications such as the ECC protocols.

All the existing fully homomorphic encryption schemes are based on three different problems, namely the bounded distance decoding problem over ideal lattice, the approximate greatest common divisor problem over integers, and the learning with error problem. In this paper, we unify the first two families of problems by introducing a new class of problems, which can be reduced from both problems. Based on this new problem, namely the bounded distance decoding over hidden ideal lattice, we present a new fully homomorphic encryption scheme. Since it is a combination of the two problems to some extent, the performance of our scheme lies between the ideal lattice based schemes and the integer based schemes. Furthermore, we also show a lower bound and upper bound of the problem that our scheme is based on. Assuming this security conjecture holds, we can incorporate smaller parameters, which will result in a scheme that is more efficient than both lattice based and integer based schemes. Hence, our scheme makes a perfect alternative to the state-of-art ring learning with error based schemes.