Improved Fully Homomorphic Encryption without Bootstrapping

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 enciphering function. In this paper I propose the improved fully homomorphic public-key encryption scheme on non-associative octonion ring over finite field without bootstrapping technique. The plaintext p consists of two sub-plaintext u and v. The proposed fully homomorphic public-key encryption scheme is immune from the “p and -p attack”. The cipher text consists of three sub-cipher texts. 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

Symmetric Ciphers for Fully Homomorphic Encryption

Fully homomorphic encryption is the latest addition to the world of cryptography. It is a type of encryption that allows operations to be done on ciphertexts, which is not possible with traditional encryption. The field has gained a lot of traction since it was first theoretically proved possible in 2009. This thesis goes through how fully homomorphic encryption works, from making a somewhat homomorphic encryption scheme, into a fully homomorphic scheme. We also explain in detail the different aspects required, such as bootstrapping and noise. Since 2009 several schemes and libraries to optimize homomorphic encryption have been suggested, so that it one day may be feasible to implement it in regular modern-day applications. Some libraries target regular developers without an extensive cryptographic background, so they may still be able to use homomorphic encryption in applications, while others aim for researchers to implement and discover the possibilities that come with fully homomorphic encryption. With a focus on the use for fully homomorphic encryption within cloud computing, this thesis focuses on how symmetric ciphers can make fully homomorphic encryption possible, also for use with small IoT devices. We look at several such ciphers that have been suggested and focus on the family of stream ciphers called Rasta. We have implemented one variant of the Rasta cipher using the software library HElib and timed its performance.Masteroppgave i informatikkINF399MAMN-INFMAMN-PRO

Integer-based fully homomorphic encryption

The concept of fully homomorphic encryption has been considered the holy grail of cryptography since the discovery of secure public key cryptography in the 1970s. Fully homomorphic encryption allows arbitrary computation on encrypted data to be performed securely. Craig Gentry\u27s new method of bootstrapping introduced in 2009 provides a technique for constructing fully homomorphic cryptosystems. In this paper we explore one such bootstrappable system based on simple integer arithmetic in a manner that someone without a high level of experience in homomorphic encryption can readily understand. Further, we present an implementation of the system as well as a lattice- based attack. We present performance results of our implementation under various parameter choices and the resistance of the system to the lattice-based attack under those parameters. Unfortunately, while the system is very interesting from a theoretical point of view, the results show that it is still not feasible for use

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

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

Cryptanalysis of Chosen Symmetric Homomorphic Schemes

Since Gentry’s breakthrough result was introduced in the year 2009, the homomorphic encryption has become a very popular topic. The main contribution of Gentry’s thesis was, that it has proven, that it actually is possible to design a fully homomorphic encryption scheme. However ground-breaking Gentry’s result was, the designs, that employ the bootstrapping technique suffer from terrible performance both in key generation and homomorphic evaluation of circuits. Some authors tried to design schemes, that could evaluate homomorphic circuits of arbitrarily many inputs without need of bootstrapping. This paper introduces notion of symmetric homomorphic encryption, analyses the security of four such proposals, published in three different papers. Our result is a known plaintext key-recovery attack on every one of these schemes

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

On FHE without bootstrapping

We investigate the use of multivariate polynomials in constructing a fully homomorphic encryption. In this work we come up with two fully homomorphic schemes. First, we propose an IND-CPA secure symmetric key homomorphic encryption scheme using multivariate polynomial ring over finite fields. This scheme gives a method of constructing a CPA secure homomorphic encryption scheme from another symmetric deterministic CPA secure scheme. We base the security of the scheme on pseudo random functions and also construct an information theoretically secure variant, rather than basing security on hard problems like Ideal Membership and Gröbner basis as seen in most polly cracker based schemes which also use multivariate polynomial rings. This scheme is not compact but has many interesting properties- It can evaluate circuits of arbitrary depths without bootstrapping for bounded length input to the algorithm. Second what follows naturally is, an attempt to make it compact we propose some changes to the scheme and analyse the scheme in (Albrecht et. al. Asiacrypt-2011). We try to make it compact but fail and realise that this could give us a Multi Party Computation protocol. Realising that polynomials leads us to non compact schemes we move propose schemes based on matrices. We then propose our candidate for a fully homomorphic encryption without bootstrapping