618 research outputs found
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
Solving multivariate polynomial systems and an invariant from commutative algebra
The complexity of computing the solutions of a system of multivariate
polynomial equations by means of Gr\"obner bases computations is upper bounded
by a function of the solving degree. In this paper, we discuss how to
rigorously estimate the solving degree of a system, focusing on systems arising
within public-key cryptography. In particular, we show that it is upper bounded
by, and often equal to, the Castelnuovo Mumford regularity of the ideal
generated by the homogenization of the equations of the system, or by the
equations themselves in case they are homogeneous. We discuss the underlying
commutative algebra and clarify under which assumptions the commonly used
results hold. In particular, we discuss the assumption of being in generic
coordinates (often required for bounds obtained following this type of
approach) and prove that systems that contain the field equations or their fake
Weil descent are in generic coordinates. We also compare the notion of solving
degree with that of degree of regularity, which is commonly used in the
literature. We complement the paper with some examples of bounds obtained
following the strategy that we describe
Public Key Encryption Supporting Plaintext Equality Test and User-Specified Authorization
In this paper we investigate a category of public key encryption schemes which supports plaintext equality test and user-specified authorization. With this new primitive, two users, who possess their own public/private key pairs, can issue token(s) to a proxy to authorize it to perform plaintext equality test from their ciphertexts. We provide a formal formulation for this primitive, and present a construction with provable security in our security model. To mitigate the risks against the semi-trusted proxies, we enhance the proposed cryptosystem by integrating the concept of computational client puzzles. As a showcase, we construct a secure personal health record application based on this primitive
Aperiodic logarithmic signatures
In this paper we propose a method to construct logarithmic signatures which
are not amalgamated transversal and further do not even have a periodic block.
The latter property was crucial for the successful attack on the system MST3 by
Blackburn et al. [1]. The idea for our construction is based on the theory in
Szab\'o's book about group factorizations [12]
Folding Alternant and Goppa Codes with Non-Trivial Automorphism Groups
The main practical limitation of the McEliece public-key encryption scheme is
probably the size of its key. A famous trend to overcome this issue is to focus
on subclasses of alternant/Goppa codes with a non trivial automorphism group.
Such codes display then symmetries allowing compact parity-check or generator
matrices. For instance, a key-reduction is obtained by taking quasi-cyclic (QC)
or quasi-dyadic (QD) alternant/Goppa codes. We show that the use of such
symmetric alternant/Goppa codes in cryptography introduces a fundamental
weakness. It is indeed possible to reduce the key-recovery on the original
symmetric public-code to the key-recovery on a (much) smaller code that has not
anymore symmetries. This result is obtained thanks to a new operation on codes
called folding that exploits the knowledge of the automorphism group. This
operation consists in adding the coordinates of codewords which belong to the
same orbit under the action of the automorphism group. The advantage is
twofold: the reduction factor can be as large as the size of the orbits, and it
preserves a fundamental property: folding the dual of an alternant (resp.
Goppa) code provides the dual of an alternant (resp. Goppa) code. A key point
is to show that all the existing constructions of alternant/Goppa codes with
symmetries follow a common principal of taking codes whose support is globally
invariant under the action of affine transformations (by building upon prior
works of T. Berger and A. D{\"{u}}r). This enables not only to present a
unified view but also to generalize the construction of QC, QD and even
quasi-monoidic (QM) Goppa codes. All in all, our results can be harnessed to
boost up any key-recovery attack on McEliece systems based on symmetric
alternant or Goppa codes, and in particular algebraic attacks.Comment: 19 page
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