32 research outputs found
A Smart Approach for GPT Cryptosystem Based on Rank Codes
The concept of Public- key cryptosystem was innovated by McEliece's
cryptosystem. The public key cryptosystem based on rank codes was presented in
1991 by Gabidulin -Paramonov-Trejtakov(GPT). The use of rank codes in
cryptographic applications is advantageous since it is practically impossible
to utilize combinatoric decoding. This has enabled using public keys of a
smaller size. Respective structural attacks against this system were proposed
by Gibson and recently by Overbeck. Overbeck's attacks break many versions of
the GPT cryptosystem and are turned out to be either polynomial or exponential
depending on parameters of the cryptosystem. In this paper, we introduce a new
approach, called the Smart approach, which is based on a proper choice of the
distortion matrix X. The Smart approach allows for withstanding all known
attacks even if the column scrambler matrix P over the base field Fq.Comment: 5 pages. to appear in Proceedings of IEEE ISIT201
On improving security of GPT cryptosystems
The public key cryptosystem based on rank error correcting codes (the GPT
cryptosystem) was proposed in 1991. Use of rank codes in cryptographic
applications is advantageous since it is practically impossible to utilize
combinatoric decoding. This enabled using public keys of a smaller size.
Several attacks against this system were published, including Gibson's attacks
and more recently Overbeck's attacks. A few modifications were proposed
withstanding Gibson's attack but at least one of them was broken by the
stronger attacks by Overbeck. A tool to prevent Overbeck's attack is presented
in [12]. In this paper, we apply this approach to other variants of the GPT
cryptosystem.Comment: 5 pages. submitted ISIT 2009.Processed on IEEE ISIT201
Polynomial-Time Key Recovery Attack on the Faure-Loidreau Scheme based on Gabidulin Codes
Encryption schemes based on the rank metric lead to small public key sizes of
order of few thousands bytes which represents a very attractive feature
compared to Hamming metric-based encryption schemes where public key sizes are
of order of hundreds of thousands bytes even with additional structures like
the cyclicity. The main tool for building public key encryption schemes in rank
metric is the McEliece encryption setting used with the family of Gabidulin
codes. Since the original scheme proposed in 1991 by Gabidulin, Paramonov and
Tretjakov, many systems have been proposed based on different masking
techniques for Gabidulin codes. Nevertheless, over the years all these systems
were attacked essentially by the use of an attack proposed by Overbeck.
In 2005 Faure and Loidreau designed a rank-metric encryption scheme which was
not in the McEliece setting. The scheme is very efficient, with small public
keys of size a few kiloBytes and with security closely related to the
linearized polynomial reconstruction problem which corresponds to the decoding
problem of Gabidulin codes. The structure of the scheme differs considerably
from the classical McEliece setting and until our work, the scheme had never
been attacked. We show in this article that this scheme like other schemes
based on Gabidulin codes, is also vulnerable to a polynomial-time attack that
recovers the private key by applying Overbeck's attack on an appropriate public
code. As an example we break concrete proposed bits security parameters in
a few seconds.Comment: To appear in Designs, Codes and Cryptography Journa
Injective Rank Metric Trapdoor Functions with Homogeneous Errors
In rank-metric cryptography, a vector from a finite dimensional linear space
over a finite field is viewed as the linear space spanned by its entries. The
rank decoding problem which is the analogue of the problem of decoding a random
linear code consists in recovering a basis of a random noise vector that was
used to perturb a set of random linear equations sharing a secret solution.
Assuming the intractability of this problem, we introduce a new construction of
injective one-way trapdoor functions. Our solution departs from the frequent
way of building public key primitives from error-correcting codes where, to
establish the security, ad hoc assumptions about a hidden structure are made.
Our method produces a hard-to-distinguish linear code together with low weight
vectors which constitute the secret that helps recover the inputs.The key idea
is to focus on trapdoor functions that take sufficiently enough input vectors
sharing the same support. Applying then the error correcting algorithm designed
for Low Rank Parity Check (LRPC) codes, we obtain an inverting algorithm that
recovers the inputs with overwhelming probability
An extension of Overbeck's attack with an application to cryptanalysis of Twisted Gabidulin-based schemes
In the present article, we discuss the decoding of Gabidulin and related
codes from a cryptographic perspective and we observe that these codes can be
decoded with the single knowledge of a generator matrix. Then, we extend and
revisit Gibson's and Overbeck's attacks on the generalised GPT encryption
scheme (instantiated with Gabidulin codes) for various ranks of the distortion
matrix and apply our attack to the case of an instantiation with twisted
Gabidulin codes