postquantum cryptography and Fully Homomorphic Encryption (FHE), respectively. A lattice can be seen as a vector space generated by all linear combinations with integer coefficients of a set R = " rv0, f, rvn - 1 ,, with rvi ! R m, of linearly independent vectors, as defined in (46); the rank of the lattice is n, and its dimension is m. Two vectors in span ^ R h are congruent if their difference is in L ^ R h . L ^R h = ) / z i rvi : z i ! Z 3 n-1 (46) i=0 Each basis is associated with the parallelepiped: P ^ R h = ) / w i rvi : w i ! ` - 1 , 1 B3 2 2 i=0 n-1 (47) v ! L^ Rh v + xv ! span ^ R h, where w For any point yv = w and xv ! P ^ Rh, the reduction in the yv modulo P ^ R h is defined as xv = yv mod P ^ Rh . The modular reduction has a different meaning for each basis, since they are associated with different parallelepipeds. An example of this is featured in Fig. 25, where the point cv is reduced modulo both P ^ Rh and P ^ Bh, producing two different points, which are represented as triangles, while L ^ Rh = L ^ Bh . Lattice Based Cryptography (LBC) is supported by the Closest Vector Problem (CVP). This problem consists of a given base R ! R n # m, and yv ! R m, finding xv ! L ^ Rh such that < yv - xv < = min zv ! L^ Bh < yv - zv <. The private basis is produced as a rotated nearly orthogonal basis, such that Babai's round-off [146] provides accurate solutions to the CVP. Rose's cryptosystem uses bases of an Optimal Hermite Normal Form (OHNF) as the public key, a subclass of Hermite Normal Forms (HNFs), where all but the first column are trivial. The decryption algorithm is modified for its implementation v -1@ ( 6 $ @ denotes roundwith the RNS. The operation 6cR ing to the nearest integer) can be replaced by the apv -1@ through an RNS Montgomery proximation nv of 6ccR reduction [139], where cv = ^c, 0, f, 0 h and the scaling by c enables the detection and correction of the errors resulting from the approximate RNS Montgomery reducv -1@ is rewritten using integer arithmetic as in tion. 6ccR t = R -1 d is an integer and d = det ^ R h . (48), where R v -1@ = 6ccR vt v t - ccR ccR d d (48) It is shown that the usage of RNS enables parallelizing the decryption in Rose's cryptosystems to significantly speed up its computation in both CPUs and GPUs [147]. Homomorphic encryption allows performing computations directly on ciphertexts, generating encrypted results as if the operations had been performed on the plaintext and then encrypted. FHE provides malleable ciphertexts, such that given two ciphertexts representing FIRST QUARTER 2021 the operands, it will be possible to produce a ciphertext encrypting its product or sum [166]. Structured lattices underpin classes of cryptosystems supporting FHE [148]. There is noise associated with the ciphertexts that grows as homomorphic operations are applied; thus, bootstrapping has been proposed, a technique in which ciphertexts are homomorphically decrypted [148]. Modern FHE systems rely on Ring Learning with Errors (RLWE), for which techniques have been proposed that limit the need for bootstrapping [147]. Batching [149] improves the performance of FHE based on the CRT by allowing multiple bits to be encrypted in a single ciphertext so that one can carry out AND and XOR sequences of bits using a single homomorphic multiplication or addition. For example, in an RLWE cryptosystem, binary polynomials are homomorphically processed in a cyclotomic ring. By noticing that certain cyclotomic polynomials factor modulo two onto a set of polynomials with the same degree, one may take advantage of the CRT to associate a plaintext bit with each one of these polynomials. Homomorphic additions and multiplications then add and multiply the bits modulo their respective polynomial, achieving coefficientwise XOR and AND operations. Rotations of these bits may be accomplished with [150]. The operations that arise from FHE are evaluated, and efficient algorithm-hiding systems are designed for applications that take advantage of those operations in S b0 S c S S c mod P (B ) S b1 = r1 S r0 S P (R ) c mod P (R ) P (B ) v 0, b v 1 of the same lattice, Figure 25. Two basis vr0, vr1 and b along with the corresponding parallelepipeds in red and grey, v is reduced modulo the two parallelepiare represented; c peds [147]. IEEE CIRCUITS AND SYSTEMS MAGAZINE 35

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