6,530 research outputs found
General Impossibility of Group Homomorphic Encryption in the Quantum World
Group homomorphic encryption represents one of the most important building
blocks in modern cryptography. It forms the basis of widely-used, more
sophisticated primitives, such as CCA2-secure encryption or secure multiparty
computation. Unfortunately, recent advances in quantum computation show that
many of the existing schemes completely break down once quantum computers reach
maturity (mainly due to Shor's algorithm). This leads to the challenge of
constructing quantum-resistant group homomorphic cryptosystems.
In this work, we prove the general impossibility of (abelian) group
homomorphic encryption in the presence of quantum adversaries, when assuming
the IND-CPA security notion as the minimal security requirement. To this end,
we prove a new result on the probability of sampling generating sets of finite
(sub-)groups if sampling is done with respect to an arbitrary, unknown
distribution. Finally, we provide a sufficient condition on homomorphic
encryption schemes for our quantum attack to work and discuss its
satisfiability in non-group homomorphic cases. The impact of our results on
recent fully homomorphic encryption schemes poses itself as an open question.Comment: 20 pages, 2 figures, conferenc
Conditionals in Homomorphic Encryption and Machine Learning Applications
Homomorphic encryption aims at allowing computations on encrypted data
without decryption other than that of the final result. This could provide an
elegant solution to the issue of privacy preservation in data-based
applications, such as those using machine learning, but several open issues
hamper this plan. In this work we assess the possibility for homomorphic
encryption to fully implement its program without relying on other techniques,
such as multiparty computation (SMPC), which may be impossible in many use
cases (for instance due to the high level of communication required). We
proceed in two steps: i) on the basis of the structured program theorem
(Bohm-Jacopini theorem) we identify the relevant minimal set of operations
homomorphic encryption must be able to perform to implement any algorithm; and
ii) we analyse the possibility to solve -- and propose an implementation for --
the most fundamentally relevant issue as it emerges from our analysis, that is,
the implementation of conditionals (requiring comparison and selection/jump
operations). We show how this issue clashes with the fundamental requirements
of homomorphic encryption and could represent a drawback for its use as a
complete solution for privacy preservation in data-based applications, in
particular machine learning ones. Our approach for comparisons is novel and
entirely embedded in homomorphic encryption, while previous studies relied on
other techniques, such as SMPC, demanding high level of communication among
parties, and decryption of intermediate results from data-owners. Our protocol
is also provably safe (sharing the same safety as the homomorphic encryption
schemes), differently from other techniques such as
Order-Preserving/Revealing-Encryption (OPE/ORE).Comment: 14 pages, 1 figure, corrected typos, added introductory pedagogical
section on polynomial approximatio
Homomorphic Encryption and Cryptanalysis of Lattice Cryptography
The vast amount of personal data being collected and analyzed through internet connected devices is vulnerable to theft and misuse. Modern cryptography presents several powerful techniques that can help to solve the puzzle of how to harness data for use while at the same time protecting it---one such technique is homomorphic encryption that allows computations to be done on data while it is still encrypted. The question of security for homomorphic encryption relates to the broader field of lattice cryptography. Lattice cryptography is one of the main areas of cryptography that promises to be secure even against quantum computing.
In this dissertation, we will touch on several aspects of homomorphic encryption and its security based on lattice cryptography. Our main contributions are:
1. proving some heuristics that are used in major results in the literature for controlling the error size in bootstrapping for fully homomorphic encryption,
2. presenting a new fully homomorphic encryption scheme that supports k-bit arbitrary operations and achieves an asymptotic ciphertext expansion of one,
3. thoroughly studying certain attacks against the Ring Learning with Errors problem,
4. precisely characterizing the performance of an algorithm for solving the Approximate Common Divisor problem
Improved Fully Homomorphic Encryption with Composite Number Modulus
Gentry’s bootstrapping technique is the most famous method of obtaining fully homomorphic encryption. In previous work I proposed a fully homomorphic encryption without bootstrapping which has the weak point in the plaintext. I also proposed a fully homomorphic encryption with composite number modulus which avoids the weak point by adopting the plaintext including the random numbers in it. In this paper I propose another fully homomorphic encryption with composite number modulus where the complexity required for enciphering and deciphering is smaller than the same modulus RSA scheme. In the proposed scheme it is proved that if there exists the PPT algorithm that decrypts the plaintext from the any ciphertexts of the proposed scheme, there exists the PPT algorithm that factors the given composite number modulus. In addition it is said that the proposed fully homomorphic encryption scheme is immune from the “p and -p attack”. Since the scheme is based on computational difficulty to solve the multivariate algebraic equations of high degree while the almost all multivariate cryptosystems proposed until now are based on the quadratic equations avoiding the explosion of the coefficients. Because proposed fully homomorphic encryption scheme is based on multivariate algebraic equations with high degree or too many variables, it is against the Gröbner basis attack, the differential attack, rank attack and so on
Fully Homomorphic Encryption with Composite Number Modulus
Gentry’s bootstrapping technique is the most famous method of obtaining fully homomorphic encryption. In previous work I proposed a fully homomorphic encryption without bootstrapping which has the weak point in the plaintext. In this paper I propose 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. The proposed fully homomorphic encryption scheme is immune from the “p and -p attack”. As the scheme is based on computational difficulty to solve the multivariate algebraic equations of high degree while the almost all multivariate cryptosystems proposed until now are based on the quadratic equations avoiding the explosion of the coefficients. Because proposed fully homomorphic encryption scheme is based on multivariate algebraic equations with high degree or too many variables, it is against the Gröbner basis attack, the differential attack, rank attack and so on.
It is proved that if there exists the PPT algorithm that decrypts the plaintext from the ciphertexts of the proposed scheme, there exists the PPT algorithm that factors the given composite number modulus
A Survey on Homomorphic Encryption Schemes: Theory and Implementation
Legacy encryption systems depend on sharing a key (public or private) among
the peers involved in exchanging an encrypted message. However, this approach
poses privacy concerns. Especially with popular cloud services, the control
over the privacy of the sensitive data is lost. Even when the keys are not
shared, the encrypted material is shared with a third party that does not
necessarily need to access the content. Moreover, untrusted servers, providers,
and cloud operators can keep identifying elements of users long after users end
the relationship with the services. Indeed, Homomorphic Encryption (HE), a
special kind of encryption scheme, can address these concerns as it allows any
third party to operate on the encrypted data without decrypting it in advance.
Although this extremely useful feature of the HE scheme has been known for over
30 years, the first plausible and achievable Fully Homomorphic Encryption (FHE)
scheme, which allows any computable function to perform on the encrypted data,
was introduced by Craig Gentry in 2009. Even though this was a major
achievement, different implementations so far demonstrated that FHE still needs
to be improved significantly to be practical on every platform. First, we
present the basics of HE and the details of the well-known Partially
Homomorphic Encryption (PHE) and Somewhat Homomorphic Encryption (SWHE), which
are important pillars of achieving FHE. Then, the main FHE families, which have
become the base for the other follow-up FHE schemes are presented. Furthermore,
the implementations and recent improvements in Gentry-type FHE schemes are also
surveyed. Finally, further research directions are discussed. This survey is
intended to give a clear knowledge and foundation to researchers and
practitioners interested in knowing, applying, as well as extending the state
of the art HE, PHE, SWHE, and FHE systems.Comment: - Updated. (October 6, 2017) - This paper is an early draft of the
survey that is being submitted to ACM CSUR and has been uploaded to arXiv for
feedback from stakeholder
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