In theory, Fully Homomorphic Encryption schemes allow users to
compute any operation over encrypted data. However in practice, one of
the major difficulties lies into determining secure cryptographic
parameters that minimize the computational cost of evaluating a circuit.
In this paper, we propose a solution to solve this open problem. Even
though it mainly focuses on TFHE, the method is generic enough to be
adapted to all the current FHE schemes.

TFHE is particularly suited, for small precision messages, from Boolean
to 5-bit integers. It is possible to instantiate bigger integers with
this scheme, however the computational cost quickly becomes unpractical.
By studying the parameter optimization problem for TFHE, we observed
that if one wants to evaluate operations on larger integers, the best
way to do it is by encrypting the message into several ciphertexts,
instead of considering bigger parameters for a single ciphertext. In the
literature, one can find some constructions going in that direction,
which are mainly based on radix and CRT representations of the message.
However, they still present some limitations, such as inefficient
algorithms to evaluate generic homomorphic lookup tables and no solution
to work with arbitrary modulus for the message space. We overcome these
limitations by proposing two new ways to evaluate homomorphic modular
reductions for any modulo in the radix approach, by introducing on the
one hand a new hybrid representation, and on the other hand by
exploiting a new efficient algorithm to evaluate generic lookup tables
on several ciphertexts. The latter is not only a programmable
bootstrapping but does not require any padding bit, as needed in the
original TFHE bootstrapping. We additionally provide benchmarks to
support our results in practice. Finally, we formalize the parameter
selection as an optimization problem, and we introduce a framework based
on it enabling easy and efficient translation of an arithmetic circuit
into an FHE graph of operation along with its optimal set of
cryptographic parameters. This framework offers a plethora of features:
fair comparisons between FHE operators, study of contexts that are
favorable to a given FHE strategy/algorithm, failure probability
selection for the entire use case, and so on.