An important problem of modern cryptography concerns secret public-key computations in algebraic structures. We construct homomorphic cryptosystems being (secret) epimorphisms f : G → H, where G,H are (publically known) groups and H is finite. A letter of a message to be encrypted is an element h ∈ H, while its encryption g ∈ G is such that f(g) = h. A homomorphic cryptosystem allows one to per...

In this paper we present adaptive key recovery attacks on NTRU-based somewhat homomorphic encryption schemes. Among such schemes, we study the proposal by Bos et al [BLLN13] in 2013. Given access to a decryption oracle, the attack allows us to compute the private key for all parameter choices. Such attacks show that one must be very careful about the use of homomorphic encryption in practice. T...

This paper introduces two designs to enhance the Boneh and Freemans linearly homomorphic signature over binary fields, to overcome the limitations to implement homomorphic signatures to the real world scenario due to the heavy calculation and under multiple signers setting for a message. Based on our concurrent work on classification on lattice-based trapdoor functions in SCIS 2017, we modify s...

We propose a general framework to develop fully homomorphic encryption schemes (FHE) without using Gentry’s technique. Initially, a private-key cryptosystem is built over Zn (n being an RSA modulus). An encryption of x ∈ Zn is a randomly chosen vector e such that Φ(e) = x where Φ is a secret multivariate polynomial. This private-key cryptosystem is not homomorphic in the sense that the vector s...

We present the homomorphic evaluation of the Prince block cipher. Our leveled implementation is based on a generalization of NTRU. We are motivated by the drastic bandwidth savings that may be achieved by scheme conversion. To unlock this advantage we turn to lightweight ciphers such as Prince. These ciphers were designed from scratch to yield fast and compact implementations on resource-constr...

We describe two improvements to Gentry's fully homomorphic scheme based on ideal lattices and its analysis: we provide a more aggressive analysis of one of the hardness assumptions (the one related to the Sparse Subset Sum Problem) and we introduce a probabilistic decryption algorithm that can be implemented with an algebraic circuit of low multiplicative degree. Combined together, these improv...

We present a general framework for developing and analyzing homomorphic cryptosystems whose security relies on the difficulty of solving systems of nonlinear equations over Zn, n being an RSA modulus. In this framework, many homomorphic cryptosystems can be conceptualized. Based on symmetry considerations, we propose a general assumption that ensures the security of these schemes. To highlight ...

Let A be a C∗-algebra and E : A → A a conditional expectation. The Kadison-Schwarz inequality for completely positive maps, E(x)∗E(x) ≤ E(x∗x), implies that ‖E(x)‖ ≤ ‖E(x∗x)‖ . In this note we show that E is homomorphic (in the sense that E(xy) = E(x)E(y) for every x, y in A) if and only if ‖E(x)‖ = ‖E(x∗x)‖ , for every x in A. We also prove that a homomorphic conditional expectation on a commu...

We will look into privacy preserving biometrics using the example of a fingerprint reader and partially homomorphic encryption. Therefore we will cover the basics necessary to understand the discussed subject, partially homomorphic encryption and fingerprint based authentication, as well as showing a concrete protocol and its implications on performance and security of the system. While securit...

After the trials of remote internet voting for local elections in 2011 and parliamentary elections in 2013, a number of local referendums has renewed interest in internet voting in Norway. The voting scheme used in Norway is not quantum-safe and it has limited voter verifiability. In this case study, we consider how we can use fully homomorphic encryption to construct a quantum-safe voting sche...