HEAAN (Homomorphic Encryption for Arithmetic of Approximate Numbers) is an open source homomorphic encryption (HE) library which implements an approximate HE scheme proposed by Cheon, Kim, Kim and Song (CKKS).[1] The first version of HEAAN was published on GitHub[2] on 15 May 2016, and later a new version of HEAAN with a bootstrapping algorithm[3] was released. Currently, the latest version is Version 2.1.[4][verification needed]

HEAAN
Developer(s)Cryptography LAB in Seoul National University
Initial releaseMay 15, 2016; 8 years ago (2016-05-15)
Repository
Written inC++
TypeHomomorphic encryption
LicenseCC BY-NC 3.0

CKKS plaintext space

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Unlike other HE schemes, 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

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Given a plaintext vector   and a scaling factor  , the plaintext vector is encoded as a polynomial   by computing   where   denotes the coefficient-wise rounding function.

Decoding method

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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 following hold:

  •  ,
  •  ,

where   denotes the Hadamard product of the same-length vectors. These properties guarantee the approximate correctness of the computations in the encoded state when the scaling factor   is chosen appropriately.

Algorithms

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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

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The key generation algorithm is following:

  • Sample a secret polynomial  .
  • Sample   (resp.  ) uniform randomly from   (resp.  ), and  .
  • Output a secret key  , a public key  , and an evaluation key  .

Encryption

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The encryption algorithm is following:

  • Sample an ephemeral secret polynomial  .
  • For a given message polynomial  , output a ciphertext  .

Decryption

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The decryption algorithm is following:

  • For a given ciphertext  , output a message    .

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. Basic homomorphic operations, addition and multiplication, are done as follows.

Homomorphic addition

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The homomorphic addition algorithm is following:

  • Given two ciphertexts   and   in  , output  .

The correctness holds as  .

Homomorphic multiplication

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The homomorphic multiplication algorithm is following:

  • Given two ciphertext   and   in  , compute    . Output  .

The correctness holds as  .

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.[5] In case of HEAAN, the modulus-switching procedure is called rescaling. The Rescaling algorithm is very simple compared to Brakerski-Gentry-Vaikuntanathan's original algorithm. Applying the rescaling algorithm after a homomomorphic multiplication, the approximation error grows linearly, not exponentially.

Rescaling

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The rescaling algorithm is following:

  • Given a ciphertext   and a new modulus  , output a rescaled ciphertext  .

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

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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.[6]

Library

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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 latest version, the multiplication of ring elements in   was accelerated by utilizing fast Fourier transform (FFT)-optimized number theoretic transform (NTT) implementation.

References

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  1. ^ Cheon, Jung Hee; Kim, Andrey; Kim, Miran; Song, Yongsoo (2017). "Homomorphic encryption for arithmetic of approximate numbers". Takagi T., Peyrin T. (eds) Advances in Cryptology – ASIACRYPT 2017. ASIACRYPT 2017. Springer, Cham. pp. 409–437. doi:10.1007/978-3-319-70694-8_15.
  2. ^ Andrey Kim; Kyoohyung Han; Miran Kim; Yongsoo Song. "An approximate HE library HEAAN". Retrieved 15 May 2016.
  3. ^ Jung Hee Cheon, Kyoohyung Han, Andrey Kim, Miran Kim and Yongsoo Song. Bootstrapping for Approximate Homomorphic Encryption. In EUROCRYPT 2018(springer).
  4. ^ snucrypto/HEAAN, Cryptography LAB in Seoul National University, 2021-07-19, retrieved 2021-07-20
  5. ^ Z. Brakerski, C. Gentry, and V. Vaikuntanathan. Fully Homomorphic Encryption without Bootstrapping. In ITCS 2012
  6. ^ Martin Albrecht. Security Estimates for the Learning with Errors Problem, https://bitbucket.org/malb/lwe-estimator