a590003

mod q l . A polynomial a ∈ A q is represented as the (l + 1) × φ(m) matrix of its evaluation at the
roots of Φ m (X) modulo p i for i = 0, . . . , l:
(
)
DoubleCRT l (a) = a(ζ j i ) mod p i
.
0≤i≤l, j∈Z ∗ m
Addition and multiplication in A q can be computed as component-wise addition and multiplication
of the entries in the two tables (modulo the appropriate primes p i ),
DoubleCRT l (a + b) = DoubleCRT l (a) + DoubleCRT l (b),
DoubleCRT l (a · b) = DoubleCRT l (a) · DoubleCRT l (b).
Also, for an element of the Galois group κ ∈ Gal, mapping a(X) ∈ A to a(X k ) mod Φ m (X), we can
evaluate κ(a) on the double-CRT representation of a just by permuting the columns in the matrix,
sending each column j to column j · k mod m.
1.3 Modules in our Library
Very roughly, our HE library consists of four layers: in the bottom layer we have modules for
implementing mathematical structures and various other utilities, the second layer implements
our Double-CRT representation of polynomials, the third layer implements the cryptosystem itself
(with the “native” plaintext space of binary polynomials), and the top layer provides interfaces
for using the cryptosystem to operate on arrays of plaintext values (using the plaintext slots as
described in Section 1.1). We think of the bottom two layers as the “math layers”, and the top
two layers as the “crypto layers”, and describe then in detail in Sections 2 and 3, respectively.
A block-diagram description of the library is given in Figure 1. Roughly, the modules NumbTh,
timing, bluestein, PAlgebra, PAlgebraModTwo, PAlgebraMod2r, Cmodulus, IndexSet and IndexMap
belong to the bottom layer, FHEcontext, SingleCRT and DoubleCRT belong to the second layer,
FHE, Ctxt and KeySwitching are in the third layer, and EncryptedArray and EncryptedArrayMod2r
are in the top layer.
2 The Math Layers
2.1 The timing module
This module contains some utility function for measuring the time that various methods take to
execute. To use it, we insert the macro FHE TIMER START at the beginning of the method(s) that
we want to time and FHE TIMER STOP at the end, then the main program needs to call the function
setTimersOn() to activate the timers and setTimersOff() to pause them. We can have at most
one timer per method/function, and the timer is called by the same name as the function itself
(using the pre-defiend variable func ). To obtain the value of a given timer (in seconds), the
application can use the function double getTime4func(const char *fncName), and the function
printAllTimers() prints the values of all timers to the standard output.
2.2 NumbTh: Miscellaneous Utilities
This module started out as an implementation of some number-theoretic algorithms (hence the
name), but since then it grew to include many different little utility functions. For example, CRT-
3
16. Design and Implementation of a Homomorphic-Encryption Library