“HEAAN”的意思、由来-开放百科全书

The first version of HEAAN was published on github^[2] in 15 May 2016, and later a new version of HEAAN with a bootstrapping algorithm^[3]

CKKS Plaintext Space

Unlikely to the other HE scheme, the CKKS scheme supports approximate arithmetics over complex numbers (hence, real numbers).

More precisely, the plaintext space of the CKKS scheme is

for some power-of-two integer

. To deal with the complex plaintext vector efficiently, Cheon et al. proposed plaintext encoding/decoding methods which exploits a ring isomorphism

Encoding Method

Given a plaintext vector

and a scaling factor

, the plaintext vector is encoded as a polynomial

Decoding Method

Given a message polynomial

and a scaling factor

, the message polynomial is decoded to a complex vector

by computing

Here the scaling factor

enables us to control the encoding/decoding error which is occurred by the rounding process. Namely, one can obtain the approximate equation

by controlling

where

and

denote the encoding and decoding algorithm, respectively.

From the ring-isomorphic property of the mapping

, for

and

, the followings hold:

These properties guarantee the approximate correctness of the computations in the encoded state when the scaling factor

is chosen appropriately.

Algorithms

The CKKS scheme basically consists of those algorithms: key Generation, encryption, decryption, homomorphic addition and multiplication, and rescaling. For a positive integer

, let

be the quotient ring of

modulo

. Let

and

be distributions over

which output polynomials with small coefficients. These distributions, the initial modulus

, and the ring dimension

are predetermined before the key generation phase.

Key Generation

Encryption

Decryption

The decryption outputs an approximate value of the original message, i.e.,

, and the approximation error is determined by the choice of distributions

When considering homomorphic operations, the evaluation errors are also included in the approximation error.

Homomorphic Addition

Homomorphic Multiplication

Note that the approximation error (on the message) exponentially grows up on the number of homomorphic multiplications. To overcome this problem, most of HE schemes usually use a modulus-switching technique which was introduced by Brakerski, Gentry and Vaikuntanathan.^[4]

In case of HEAAN, the modulus-switching procedure is called rescaling. the Rescaling algorithm is very simple compared to Brakerski-Gentry-Vaikuntanatahn's original algorithm.

Applying the rescaling algorithm after a homomomorphic multiplication, the approximation error grows linearly, not exponentially.

Rescaling

The total procedure of the CKKS scheme is as following: Each plaintext vector

which consists of complex (or real) numbers is firstly encoded as a polynomial

by the encoding method, and then encrypted as a ciphertext

. After several homomorphic operations, the resulting ciphertext is decrypted as a polynomial

and then decoded as a plaintext vector

which is the final output.

Security

The IND-CPA security of the CKKS scheme is based on the hardness assumption of the Ring learning with errors (RLWE) problem, the ring variant of very promising lattice-based hard problem Learning with errors (LWE).

Currently the best known attacks for RLWE over a power-of-two cyclotomic ring are general LWE attacks such as dual attack and primal attack.

The bit security of the CKKS scheme based on known attacks was estimated by Albrecht's LWE estimator.^[5]

Library

Version 1.0, 1.1 and 2.1 have been released so far. Version 1.0 is the first implementation of the CKKS scheme without bootstrapping.

In the second version, the bootstrapping algorithm was attached so that users are able to address large-scale homomorphic computations.

In Version 2.1, currently the lastest version, the multiplication of ring elements in

was accelerated by utilizing fast Fourier transform (FFT)-optimized number theoretic transform (NTT) implementation.

References

1. ^{{cite conference |last1=Cheon |first1=Jung Hee |last2=Kim |first2=Andrey |last3=Kim |first3=Miran |last4=Song |first4=Yongsoo |title=Homomorphic encryption for arithmetic of approximate numbers |publisher=Springer, Cham |conference=ASIACRYPT 2017 |date=2017 |book-title=Takagi T., Peyrin T. (eds) Advances in Cryptology – ASIACRYPT 2017 |pages=409–437|doi=10.1007/978-3-319-70694-8_15 }}
2. ^{{cite web|title=An approximate HE library HEAAN|url=https://github.com/snucrypto/HEAAN|author=Andrey Kim|author2=Kyoohyung Han|author3=Miran Kim|author4=Yongsoo Song|accessdate=15 May 2016}}
3. ^Jung Hee Cheon, Kyoohyung Han, Andrey Kim, Miran Kim and Yongsoo Song. [https://eprint.iacr.org/2018/153 Bootstrapping for Approximate Homomorphic Encryption]. In EUROCRYPT 2018(springer).
4. ^Z. Brakerski, C. Gentry, and V. Vaikuntanathan. Fully Homomorphic Encryption without Bootstrapping. In ITCS 2012
5. ^Martin Albrecht. Security Estimates for the Learning with Errors Problem, https://bitbucket.org/malb/lwe-estimator