451 research outputs found
On non-abelian homomorphic public-key cryptosystems
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 element
of H, while its encryption g element of G is such that f(g)=h. A homomorphic
cryptosystem allows one to perform computations (operating in a group G) with
encrypted information (without knowing the original message over H).
In this paper certain homomorphic cryptosystems are constructed for the first
time for non-abelian groups H (earlier, homomorphic cryptosystems were known
only in the Abelian case). In fact, we present such a system for any solvable
(fixed) group H.Comment: 15 pages, LaTe
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
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
Public-key cryptography and invariant theory
Public-key cryptosystems are suggested based on invariants of groups. We give
also an overview of the known cryptosystems which involve groups.Comment: 10 pages, LaTe
Constructions in public-key cryptography over matrix groups
ISBN : 978-0-8218-4037-5International audienceThe purpose of the paper is to give new key agreement protocols (a multi-party extension of the protocol due to Anshel-Anshel-Goldfeld and a generalization of the Diffie-Hellman protocol from abelian to solvable groups) and a new homomorphic public-key cryptosystem. They rely on difficulty of the conjugacy and membership problems for subgroups of a given group. To support these and other known cryptographic schemes we present a general technique to produce a family of instances being matrix groups (over finite commutative rings) which play a role for these schemes similar to the groups in the existing cryptographic constructions like RSA or discrete logarithm
Quantum Algorithms for Some Hidden Shift Problems
Almost all of the most successful quantum algorithms discovered to date exploit the ability of the Fourier transform to recover subgroup structures of functions, especially periodicity. The fact that Fourier transforms can also be used to capture shift structure has received far less attention in the context of quantum computation. In this paper, we present three examples of "unknown shift" problems that can be solved efficiently on a quantum computer using the quantum Fourier transform. For one of these problems, the shifted Legendre symbol problem, we give evidence that the problem is hard to solve classically, by showing a reduction from breaking algebraically homomorphic cryptosystems. We also define the hidden coset problem, which generalizes the hidden shift problem and the hidden subgroup problem. This framework provides a unified way of viewing the ability of the Fourier transform to capture subgroup and shift structure
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