# A New Approach for Algebraically Homomorphic Encryption

@article{Armknecht2008ANA, title={A New Approach for Algebraically Homomorphic Encryption}, author={Frederik Armknecht and Ahmad-Reza Sadeghi}, journal={IACR Cryptol. ePrint Arch.}, year={2008}, volume={2008}, pages={422} }

The existence of an efficient and provably secure algebraically homomorphic scheme (AHS), i.e., one that supports both addition and multiplication operations, is a long stated open problem. All proposals so far are either insecure or not provable secure, inefficient, or allow only for one multiplication (and arbitrary additions). As only very limited progress has been made on the existing approaches in the recent years, the question arises whether new methods can lead to more satisfactory…

## 42 Citations

Group homomorphic encryption: characterizations, impossibility results, and applications

- Computer Science, MathematicsDes. Codes Cryptogr.
- 2013

A new cryptosystem is designed which provides features that are unique up to now: Its IND-CPA security is based on the k-linear problem introduced by Shacham, and Hofheinz and Kiltz, while its IND-CCA1 security isbased on a new k-problem that is proved to have the same progressive property.

An RNS variant of fully homomorphic encryption over integers

- Computer Science, Mathematics
- 2017

A residue number system (RNS) variant to FHE of [12], which is also applicable to the DGHV scheme and improves the original scheme’s security, since the former relies only on the hardness of the A-GCD problem and eliminates the need for the sparsesubset-sum problem used in the original MS procedure.

Enhanced Flexibility for Homomorphic Encryption Schemes via CRT

- Computer Science, Mathematics
- 2012

This paper presents a simple technique for ElGamal-type encryption schemes which facilitates encryption in an additively homomorphic manner, and shows that the CRT may be used to turn an intractable problem into a tractable one.

Royal Holloway, University of London A Survey of Fully Homomorphic Encryption

- Mathematics, Computer Science
- 2012

The theory of homomorphic encryption is discussed and several encryption schemes with such properties are reviewed, including Gentry’s construction, and a brief summary of the earlier work related to homomorphicryption is given.

Symmetric Fully Homomorphic Encryption Using Decidable Matrix Equations

- Computer Science, MathematicsSIN
- 2014

The paper gives the formal definitions of new framework for secure cloud computing -- the primitive of compact symmetric fully homomorphic encryption scheme whose security relies on the difficulty of some NP-complete problem.

Fully homomorphic encryption using ideal lattices

- Computer Science, MathematicsSTOC '09
- 2009

This work proposes a fully homomorphic encryption scheme that allows one to evaluate circuits over encrypted data without being able to decrypt, and describes a public key encryption scheme using ideal lattices that is almost bootstrappable.

Design and implementation of low-depth pairing-based homomorphic encryption scheme

- Computer Science, MathematicsJournal of Cryptographic Engineering
- 2018

The design of a scheme based on pairings and elliptic curves, that is able to handle applications where the number of multiplication is not too high, with interesting practical efficiency when compared to lattice-based solutions is focused on.

How to Combine Homomorphic Encryption and Garbled Circuits - Improved Circuits and Computing the Minimum Distance Efficiently

- Computer Science, Mathematics
- 2009

Ecient constant-round protocols for secure integer comparison, and the related problems of minimum selec- tion and minimum distance, which are crucial building blocks of many cryptographic schemes such as privacy preserving biometric authentica- tion (e.g., face recognition, ngerprint matching, etc).

Additively Homomorphic Encryption with d-Operand Multiplications

- Computer Science, MathematicsIACR Cryptol. ePrint Arch.
- 2008

This work defines a theoretical object, chained encryption schemes, which allow an efficient evaluation of polynomials of degree d over encrypted data, and proposes a chained encryption scheme whose INDCPA security is based on a worst-case/average-case reduction from uSVP.

A systematic approach to practically efficient general two-party secure function evaluation protocols and their modular design

- Computer ScienceJ. Comput. Secur.
- 2013

This article argues that in many cases the most efficient SFE protocols are obtained by combining several basic techniques, e.g., garbled circuits and additively homomorphic encryption, and presents a framework in which today's most efficient techniques for general SFE can be viewed as building blocks with well-defined interfaces that can be easily combined into a complete efficient solution.

## References

SHOWING 1-10 OF 41 REFERENCES

Multiparty Computation from Threshold Homomorphic Encryption

- Mathematics, Computer ScienceEUROCRYPT
- 2000

It is shown that given keys for any sufficiently efficient system of this type, general MPC protocols for n parties can be devised which are secure against an active adversary that corrupts any minority of the parties.

A Generalisation, a Simplification and Some Applications of Paillier's Probabilistic Public-Key System

- Mathematics, Computer SciencePublic Key Cryptography
- 2001

We propose a generalisation of Paillier's probabilistic public key system, in which the expansion factor is reduced and which allows to adjust the block length of the scheme even after the public key…

Additively Homomorphic Encryption with d-Operand Multiplications

- Computer Science, MathematicsIACR Cryptol. ePrint Arch.
- 2008

This work defines a theoretical object, chained encryption schemes, which allow an efficient evaluation of polynomials of degree d over encrypted data, and proposes a chained encryption scheme whose INDCPA security is based on a worst-case/average-case reduction from uSVP.

Non-interactive cryptocomputing for NC/sup 1/

- Computer Science, Mathematics40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)
- 1999

This paper gives a one round, and thus round optimal, protocol for secure evaluation of circuits which is in polynomial time for NC/sup 1/ circuits and suggests applications to network and mobile computing.

Secure Games with Polynomial Expressions

- Computer Science, MathematicsICALP
- 2001

This work presents the first private information retrieval (PIR) scheme which is both, deterministically correct and has poly-logarithmic communication complexity, and presents additional applications such as the "lists' intersection predicate" which is useful for secure conduct of e-commerce procedures, such as negotiation methods known as "settlement escrows" in the legal/ economics/ business literature.

A Provably Secure Additive and Multiplicative Privacy Homomorphism

- Computer Science, MathematicsISC
- 2002

This work presents one such PH (none was known so far) which can be proven secure against known-cleartext attacks, as long as the ciphertext space is much larger than the cleartext space.

Evaluating 2-DNF Formulas on Ciphertexts

- Computer Science, MathematicsTCC
- 2005

A homomorphic public key encryption scheme that allows the public evaluation of ψ given an encryption of the variables x1,...,xn and can evaluate quadratic multi-variate polynomials on ciphertexts provided the resulting value falls within a small set.

Algorithms for Black-Box Fields and their Application to Cryptography (Extended Abstract)

- Computer Science, MathematicsCRYPTO
- 1996

The results show that any algebraically homomorphic cryptosystem can be broken in sub-exponential time and it is proved that manipulating black box fields over the rationals is as hard as factoring integers.

Cryptographic Hardness Based on the Decoding of Reed–Solomon Codes

- Computer Science, MathematicsIEEE Transactions on Information Theory
- 2008

This paper deals with samplable PR instances over parameter choices for which decoding is not known to be feasibly solvable and where part of the solution polynomial is the hidden input, and puts forth a natural decisional intractability assumption that relates to this decoding problem.

ON DATA BANKS AND PRIVACY HOMOMORPHISMS

- Computer Science, Mathematics
- 1978

It appears likely that there exist encryption functions which permit encrypted data to be operated on without preliminary decryption of the operands, for many sets of interesting operations.