Design and Implementation of a Homomorphic ... - Researcher

turn, are sometimes partitioned further into a number of “digits”, corresponding to the columns in
our key-switching matrices. (See the explanation of this partition in Section 3.1.6.) These subsets
are stored in the following data members:
IndexSet ctxtPrimes; // the ciphertext primes
IndexSet specialPrimes; // the "special" primes
vector digits; // digits of ctxt/columns of key-switching matrix
The FHEcontext class provides also some convenience functions for computing the product of a
subset of small primes, as well as the “size” of that product (i.e., its logarithm), via the methods:
ZZ productOfPrimes(const IndexSet& s) const;
void productOfPrimes(ZZ& p, const IndexSet& s) const;
double logOfPrime(unsigned i) const; // = log(ithPrime(i))
double logOfProduct(const IndexSet& s) const;
Finally, the FHEcontext module includes some utility functions for adding moduli to the chain.
The method addPrime(long p, bool isSpecial) adds a single prime p (either “special” or not),
after checking that p has 2m’th roots of unity and it is not already in the list. Then we have three
higher-level functions:
double AddPrimesBySize(FHEcontext& c, double size, bool special=false);
Adds to the chain primes whose product is at least exp(size), returns the natural logarithm
of the product of all added primes.
double AddPrimesByNumber(FHEcontext& c, long n, long atLeast=1, bool special=false);
Adds n primes to the chain, all at least as large as the atLeast argument, returns the natural
logarithm of the product of all added primes.
void buildModChain(FHEcontext& c, long d, long t=3);
Build modulus chain for a circuit of depth d, using t digits in key-switching. This function
puts d ciphertext primes in the moduli vector, and then as many “special” primes as needed
to mod-switch fresh ciphertexts (see Section 3.1.6).
2.8 DoubleCRT: Efficient Polynomial Arithmetic
The heart of our library is the DoubleCRT class that manipulates polynomials in Double-CRT
representation. A DoubleCRT object is tied to a specific FHEcontext, and at any given time it is
defined relative to a subset of small primes from our list, S ⊆ [0, . . . , context.moduli.size() − 1].
Denoting the product of these small primes by q = ∏ i∈S p i, a DoubleCRT object represents a
polynomial a ∈ A q by a matrix with φ(m) columns and one row for each small prime p i (with i ∈ S).
The i’th row contains the FFT representation of a modulo p i , i.e. the evaluations {[a(ζ j
i
)] pi : j ∈
Z ∗ m}, where ζ i is some primitive m-th root of unity modulo p i .
Although the FHEcontext must remain fixed throughout, the set S of primes can change dynamically,
and so the matrix can lose some rows and add other ones as we go. We thus keep these
rows in a dynamic IndexMap data member, and the current set of indexes S is available via the
method getIndexSet(). We provide the following methods for changing the set of primes:
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