# 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} }

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.

## 34 Citations

### Fully Homomorphic Encryption Without Noise

- Mathematics, Computer ScienceResearch Anthology on Artificial Intelligence Applications in Security
- 2019

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?

- Mathematics, Computer ScienceIACR Cryptol. ePrint Arch.
- 2016

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

- Mathematics, Computer ScienceIACR Cryptol. ePrint Arch.
- 2016

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

- Computer Science, Mathematics
- 2017

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

- Mathematics, Computer ScienceIACR Cryptol. ePrint Arch.
- 2015

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

- Computer Science, MathematicsIACR Cryptol. ePrint Arch.
- 2015

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

- Mathematics, Computer ScienceIACR Cryptol. ePrint Arch.
- 2015

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

- Mathematics, Computer ScienceIACR Cryptol. ePrint Arch.
- 2015

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

- Computer Science, MathematicsACM Comput. Surv.
- 2018

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

- Computer Science, Mathematics
- 2018

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.

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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.

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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…

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