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similar methods have been restricted to small datasets (on the order of a few hundred to a
thousand elements) or data with low dimension (generally 1-4).
In this work[17], the Stanford team first constructed a working implementation of the scaleinvariant
leveled homomorphic encryption system of Brakerski. Then, by taking advantage of
batched computation as well as a message encoding technique based on the Chinese Remainder
Theorem, they showed that it becomes not only possible, but computationally feasible, to
perform statistical analysis on encrypted datasets with over four million elements and dimension
as high as 24. By using these methods along with some additional optimizations, the team was
able to demonstrate the viability of using leveled homomorphic encryption for large scale
statistical analysis.
The IBM team designed, implemented and delivered a Homomorphic Encryption (HE) software
library[18] that implements the Brakerski-Gentry-Vaikuntanathan (BGV) homomorphic
encryption scheme, along with many optimizations to make homomorphic evaluation runs faster,
focusing mostly on effective use of the Smart-Vercauteren ciphertext packing techniques. Our
library is written in C++ and uses the Number Theory Library (NTL) mathematical library. The
NTL is a high-performance, portable C++ library providing data structures and algorithms for
manipulating signed, arbitrary length integers, and for vectors, matrices, and polynomials over
the integers and over finite fields (and can be found at http://www.shoup.net/ntl).
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. 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 our work[18]. A block-diagram description of the library is given in Figure 1.
At the top level of the library we provide some interfaces that allow the application to manipulate
arrays of plaintext values homomorphically. The arrays are translated to plaintext polynomials
using the encoding/decoding routines and then encrypted and manipulated homomorphically
using the lower-level interfaces from the crypto layer.
The basic operations that we have in the HE library scheme are the usual key-generation,
encryption, and decryption, the homomorphic evaluation routines for addition, multiplication and
automorphism (and also addition-of-constant and multiplication-by-constant), and the ciphertext
maintenance operations of key-switching and modulus-switching.
In addition to the software described above the PROCEED AHEAD team has delivered many
significant publications which are summarized in the Publications section of this report.
Approved for Public Release; Distribution Unlimited.
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