Homomorphic public-key cryptosystems and encrypting boolean circuits

In this paper homomorphic cryptosystems are designed for the first time over any finite group. Applying Barrington's construction we produce for any boolean circuit of the logarithmic depth its encrypted simulation of a polynomial size over an appropriate finitely generated group

Homomorphic public-key cryptosystems and encrypting boolean circuits

International audienceGiven an arbitrary finite nontrivial group, we describe a probabilistic public-key cryptosystem in which the decryption function is chosen to be a suitable epimorphism from the free product of finite Abelian groups onto this finite group. It extends the quadratic residue cryptosystem (based on a homomorphism onto the group of two elements) due to Rabin – Goldwasser – Micali. The security of the cryptosystem relies on the intractability of factoring integers. As an immediate corollary of the main construction, we obtain a more direct proof (based on the Barrington technique) of Sander-Young-Yung result on an encrypted simulation of a boolean circuit of the logarithmic depth

RSA and redactable blockchains

A blockchain is redactable if a private key holder (e.g. a central authority) can change any single block without violating integrity of the whole blockchain, but no other party can do that. In this paper, we offer a simple method of constructing redactable blockchains inspired by the ideas underlying the well-known RSA encryption scheme. Notably, our method can be used in conjunction with any reasonable hash function that is used to build a blockchain. Public immutability of a blockchain in our construction is based on the computational hardness of the RSA problem and not on properties of the underlying hash function. Corruption resistance is based on the computational hardness of the discrete logarithm problem.Comment: 5 page

Using decision problems in public key cryptography

There are several public key establishment protocols as well as complete public key cryptosystems based on allegedly hard problems from combinatorial (semi)group theory known by now. Most of these problems are search problems, i.e., they are of the following nature: given a property P and the information that there are objects with the property P, find at least one particular object with the property P. So far, no cryptographic protocol based on a search problem in a non-commutative (semi)group has been recognized as secure enough to be a viable alternative to established protocols (such as RSA) based on commutative (semi)groups, although most of these protocols are more efficient than RSA is. In this paper, we suggest to use decision problems from combinatorial group theory as the core of a public key establishment protocol or a public key cryptosystem. By using a popular decision problem, the word problem, we design a cryptosystem with the following features: (1) Bob transmits to Alice an encrypted binary sequence which Alice decrypts correctly with probability "very close" to 1; (2) the adversary, Eve, who is granted arbitrarily high (but fixed) computational speed, cannot positively identify (at least, in theory), by using a "brute force attack", the "1" or "0" bits in Bob's binary sequence. In other words: no matter what computational speed we grant Eve at the outset, there is no guarantee that her "brute force attack" program will give a conclusive answer (or an answer which is correct with overwhelming probability) about any bit in Bob's sequence.Comment: 12 page

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

Rewriting Methods in Groups with Applications to Cryptography

In this thesis we describe how various rewriting methods in combinatorial group theory can be used to diffuse information about group elements, which makes it possible to use these techniques as an important constituent in cryptographic primitives. We also show that, while most group-based cryptographic primitives employ the complexity of search versions of algorithmic problems in group theory, it is also possible to use the complexity of decision problems, in particular the word problem, to claim security of relevant protocols