• Corpus ID: 11962791

Notes on Two Fully Homomorphic Encryption Schemes Without Bootstrapping

@article{Wang2015NotesOT,
  title={Notes on Two Fully Homomorphic Encryption Schemes Without Bootstrapping},
  author={Yongge Wang},
  journal={IACR Cryptol. ePrint Arch.},
  year={2015},
  volume={2015},
  pages={519}
}
  • Yongge Wang
  • Published 2015
  • Computer Science, Mathematics
  • IACR Cryptol. ePrint Arch.
Recently, IACR ePrint archive posted two fully homomorphic encryption schemes without bootstrapping. In this note, we show that these schemes are trivially insecure. Furthermore, we also show that the encryption schemes of Liu and Wang [6] in CCS 2012 and the encryption scheme of Liu, Bertino, and Xun [5] in ASIACCS 2014 are insecure either. 

Fully Homomorphic Encryption Without Noise

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The authors present a novel fully homomorphic encryption scheme operating between ZN and capable of arbitrarily performing additions and multiplications, which is compact and each operation performed on any two ciphertexts produces a fresh ciphertext without any associated noise.

Fully homomorphic encryption must be fat or ugly?

TLDR
It is proved that fully homomorphic schemes are impossible for several algebraic structures, and a conjecture stating that secure FHE schemes must either have a significant ciphertext expansion or use unusual algebraic structure is proposed.

Fully Homomorphic Encryption with Isotropic Elements

  • M. Yagisawa
  • Mathematics, Computer Science
    IACR Cryptol. ePrint Arch.
  • 2016
TLDR
This paper improves the previous scheme by adopting the non-zero isotropic octonions so that the “m and -m attack” is not useful because in proposed scheme many ciphertexts exist where the plaintext m is not zero and the norm is zero.

A Survey on Homomorphic Encryption Schemes: Theory and Implementation

TLDR
The basics of HE and the details of the well-known Partially Homomorphic Encryption and Somewhat HomomorphicEncryption, which are important pillars of achieving FHE, are presented and the main FHE families, which have become the base for the other follow-up FHE schemes are presented.

Fully Homomorphic Encryption with Composite Number Modulus

  • M. Yagisawa
  • Mathematics, Computer Science
    IACR Cryptol. ePrint Arch.
  • 2015
TLDR
This paper proposes the improved fully homomorphic encryption scheme on non-associative octonion ring over finite ring with composite number modulus where the plaintext p consists of three numbers u,v,w and it is proved that if there exists the PPT algorithm that decrypts the plain Text from the ciphertexts of the proposed scheme, there exist the P PT algorithm that factors the given composite numbermodulus.

Fully Homomorphic Encryption without bootstrapping

  • M. Yagisawa
  • Computer Science, Mathematics
    IACR Cryptol. ePrint Arch.
  • 2015
TLDR
This book provides the new fully homomorphic encryption scheme without “bootstrapping” which runs fast for practical use and is immune from the attacks by the quantum computers.

Noise-free Symmetric Fully Homomorphic Encryption based on noncommutative rings

TLDR
A verifiable FHE is proposed, where the receiver can check the validity of ciphertexts and the scheme is one-way secure against chosen plaintext attacks based on the factorization problem of matrices over noncommutative rings.

Fully Homomorphic Encryption on Octonion Ring

  • M. Yagisawa
  • Mathematics, Computer Science
    IACR Cryptol. ePrint Arch.
  • 2015
TLDR
The improved fully homomorphic encryption scheme on non-associative octonion ring over finite field without bootstrapping technique is proposed, based on multivariate algebraic equations with high degree or too many variables while the almost all multivariate cryptosystems proposed until now are based on the quadratic equations avoiding the explosion of the coefficients.

A Survey on Homomorphic Encryption Schemes

TLDR
The basics of HE and the details of the well-known Partially Homomorphic Encryption and Somewhat Homomorphic encryption schemes, which are important pillars for achieving FHE, are presented and the implementations and recent improvements in Gentry-type FHE schemes are surveyed.

Fully Homomorphic Public-Key Encryption Against Ciphertext Square Attack with Two Ciphertexts

TLDR
A new FHPKE against "ciphertext square attack" is proposed which does not need the bootstrapping and does not require the large complexity for enciphering and it is shown that proposed system is immune from “cipher Text square attack’, “m and -m attack” and the Grobner basis attacks.

References

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Practical Fully Homomorphic Encryption without Noise Reduction

  • Dongxi Liu
  • Computer Science, Mathematics
    IACR Cryptol. ePrint Arch.
  • 2015
TLDR
A new fully homomorphic encryption (FHE) scheme that is efficient for practical applications and proved the hardness of recovering encryption keys from any number of ciphertexts with chosen plaintexts and then the semantic security of the scheme.

Fully Homomorphic Encryption without bootstrapping

  • M. Yagisawa
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    IACR Cryptol. ePrint Arch.
  • 2015
TLDR
This book provides the new fully homomorphic encryption scheme without “bootstrapping” which runs fast for practical use and is immune from the attacks by the quantum computers.

When Homomorphism Becomes a Liability

TLDR
It is shown that an encryption scheme cannot have a simple decryption function and be homomorphic at the same time, even with added noise, and that the recent candidate fully homomorphic encryption is insecure.

Privacy of outsourced k-means clustering

TLDR
This paper presents a method that allows the data owner to encrypt its data with a homomorphic encryption scheme and the service provider to perform k-means clustering directly over the encrypted data.

Query encrypted databases practically

TLDR
It is demonstrated that aggregate SQL queries with range conditions can be performed efficiently over encrypted databases, without decrypting the databases first, by using the new homomorphic encryption scheme.

The Octonions

The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics.

On Quaternions and Octonions

This book investigates the geometry of quaternion and octonion algebras. Following a comprehensive historical introduction, the book illuminates the special properties of 3- and 4-dimensional

Quaternions and Octonions

We have all seen how useful complex numbers are for describing isometries of the plane. We will now look at the quaternionic numbers and how useful they are for describing rotations of ℝ3. Indeed

Fully homomorphic encryption without bootstrapping Cryptology ePrint Archive

  • Report
  • 2015