arXiv subjects
Publications and source records attributed to Jean-Pierre Tillich.
We propose a simple decoding algorithm for CSS codes taking into account the correlations between the X part and the Z part of the error. Applying this idea to surface codes, we derive an improved version of the perfect matching decoding algorithm which uses these X/Z correlations.
For a given support $L \in \mathbb{F}_{q^m}^n$ and a polynomial $g\in \mathbb{F}_{q^m}[x]$ with no roots in $\mathbb{F}_{q^m}$, we prove equality between the $q$-ary Goppa codes $Γ_q(L,N(g)) = Γ_q(L,N(g)/g)$ where $N(g)$ denotes the norm of $g$, that is $g^{q^{m-1}+\cdots +q+1}.$ In particular, for $m=2$, that is, for a quadratic extension, we get $Γ_q(L,g^q) = Γ_q(L,g^{q+1})$. If $g$ has roots in $\mathbb{F}_{q^m}$, then we do not necessarily have equality and we prove that the difference of the dimensions of the two codes is bounded above by the number of distinct roots of $g$ in $\mathbb{F}_{q^m}$. These identities provide numerous code equivalences and improved designed parameters for some families of classical Goppa codes.
Löndahl and Johansson proposed last year a variant of the McEliece cryptosystem which replaces Goppa codes by convolutional codes. This modification is supposed to make structural attacks more difficult since the public generator matrix of this scheme contains large parts which are generated completely at random. They proposed two schemes of this kind, one of them consists in taking a Goppa code and extending it by adding a generator matrix of a time varying convolutional code. We show here that this scheme can be successfully attacked by looking for low-weight codewords in the public code of this scheme and using it to unravel the convolutional part. It remains to break the Goppa part of this scheme which can be done in less than a day of computation in the case at hand.
The current best asymptotic lower bound on the minimum distance of quantum LDPC codes with fixed non-zero rate is logarithmic in the blocklength. We propose a construction of quantum LDPC codes with fixed non-zero rate and prove that the minimum distance grows proportionally to the square root of the blocklength.
Baldi et \textit{al.} proposed a variant of McEliece's cryptosystem. The main idea is to replace its permutation matrix by adding to it a rank 1 matrix. The motivation for this change is twofold: it would allow the use of codes that were shown to be insecure in the original McEliece's cryptosystem, and it would reduce the key size while keeping the same security against generic decoding attacks. The authors suggest to use generalized Reed-Solomon codes instead of Goppa codes. The public code built with this method is not anymore a generalized Reed-Solomon code. On the other hand, it contains a very large secret generalized Reed-Solomon code. In this paper we present an attack that is built upon a distinguisher which is able to identify elements of this secret code. The distinguisher is constructed by considering the code generated by component-wise products of codewords of the public code (the so-called "square code"). By using square-code dimension considerations, the initial generalized Reed-Solomon code can be recovered which permits to decode any ciphertext. A similar technique has already been successful for mounting an attack against a homomorphic encryption scheme suggested by Bogdanoc et \textit{al.}. This work can be viewed as another illustration of how a distinguisher of Reed-Solomon codes can be used to devise an attack on cryptosystems based on them.
Bogdanov and Lee suggested a homomorphic public-key encryption scheme based on error correcting codes. The underlying public code is a modified Reed-Solomon code obtained from inserting a zero submatrix in the Vandermonde generating matrix defining it. The columns that define this submatrix are kept secret and form a set $L$. We give here a distinguisher that detects if one or several columns belong to $L$ or not. This distinguisher is obtained by considering the code generated by component-wise products of codewords of the public code (the so called "square code"). This operation is applied to punctured versions of this square code obtained by picking a subset $I$ of the whole set of columns. It turns out that the dimension of the punctured square code is directly related to the cardinality of the intersection of $I$ with $L$. This allows an attack which recovers the full set $L$ and which can then decrypt any ciphertext.
We generalize a construction of non-binary quantum LDPC codes over $\F_{2^m}$ due to \cite{KHIS11a} and apply it in particular to toric codes. We obtain in this way not only codes with better rates than toric codes but also improve dramatically the performance of standard iterative decoding. Moreover, the new codes obtained in this fashion inherit the distance properties of the underlying toric codes and have therefore a minimum distance which grows as the square root of the length of the code for fixed $m$.
We present a generalization of the construction of graphs by Lubotzky, Phillips and Sarnak in their celebrated article "Ramanujan graphs". The new approach consists in using octonion algebras rather than quaternions. A key tool is the existing result of the unique factorization of integral octonions. The families obtained by this mean present not only the same spectral property that make them good expanders, but also show a larger girth, yielding a new record for regular graphs.
We present a theory of quantum stabilizer turbo-encoders with unbounded minimum distance. This theory is presented under a framework common to both classical and quantum turbo-encoding theory. The main conditions to have an unbounded minimum distance are that the inner seed encoder has to be recursive, and either systematic or with a totally recursive truncated decoder. This last condition has been introduced in order to obtain a theory viable in the quantum stabilizer case, since it was known that in this case the inner seed encoder could not be recursive and systematic in the same time.
This paper addresses the issue of design of low-rate sparse-graph codes with linear minimum distance in the blocklength. First, we define a necessary condition which needs to be satisfied when the linear minimum distance is to be ensured. The condition is formulated in terms of degree-1 and degree-2 variable nodes and of low-weight codewords of the underlying code, and it generalizies results known for turbo codes [8] and LDPC codes. Then, we present a new ensemble of low-rate codes, which itself is a subclass of TLDPC codes [4], [5], and which is designed under this necessary condition. The asymptotic analysis of the ensemble shows that its iterative threshold is situated close to the Shannon limit. In addition to the linear minimum distance property, it has a simple structure and enjoys a low decoding complexity and a fast convergence.
We present two new algorithms for the reconstruction of turbo codes from a noisy intercepted bitstream. With these algorithms, we were able to reconstruct various turbo codes with realistic parameter sizes. To the best of our knowledge, these are the first algorithms able to recover the whole permutation of a turbo code in the presence of high noise levels.
We cryptanalyse here two variants of the McEliece cryptosystem based on quasi-cyclic codes. Both aim at reducing the key size by restricting the public and secret generator matrices to be in quasi-cyclic form. The first variant considers subcodes of a primitive BCH code. We prove that this variant is not secure by finding and solving a linear system satisfied by the entries of the secret permutation matrix. The other variant uses quasi-cyclic low density parity-check codes. This scheme was devised to be immune against general attacks working for McEliece type cryptosystems based on low density parity-check codes by choosing in the McEliece scheme more general one-to-one mappings than permutation matrices. We suggest here a structural attack exploiting the quasi-cyclic structure of the code and a certain weakness in the choice of the linear transformations that hide the generator matrix of the code. Our analysis shows that with high probability a parity-check matrix of a punctured version of the secret code can be recovered in cubic time complexity in its length. The complete reconstruction of the secret parity-check matrix of the quasi-cyclic low density parity-check codes requires the search of codewords of low weight which can be done with about $2^{37}$ operations for the specific parameters proposed.
We present a theory of quantum serial turbo-codes, describe their iterative decoding algorithm, and study their performances numerically on a depolarization channel. Our construction offers several advantages over quantum LDPC codes. First, the Tanner graph used for decoding is free of 4-cycles that deteriorate the performances of iterative decoding. Secondly, the iterative decoder makes explicit use of the code's degeneracy. Finally, there is complete freedom in the code design in terms of length, rate, memory size, and interleaver choice. We define a quantum analogue of a state diagram that provides an efficient way to verify the properties of a quantum convolutional code, and in particular its recursiveness and the presence of catastrophic error propagation. We prove that all recursive quantum convolutional encoder have catastrophic error propagation. In our constructions, the convolutional codes have thus been chosen to be non-catastrophic and non-recursive. While the resulting families of turbo-codes have bounded minimum distance, from a pragmatic point of view the effective minimum distances of the codes that we have simulated are large enough not to degrade the iterative decoding performance up to reasonable word error rates and block sizes. With well chosen constituent convolutional codes, we observe an important reduction of the word error rate as the code length increases.
Trellises play an important theoretical and practical role for classical codes. Their main utility is to devise complexity-efficient error estimation algorithms. Here, we describe trellis representations for quantum stabilizer codes. We show that they share the same properties as their classical analogs. In particular, for any stabilizer code it is possible to find a minimal trellis representation. Our construction is illustrated by two fundamental error estimation algorithms.
The purpose of this paper is to develop a "calculus" on graphs that allows graph theory to have new connections to analysis. For example, our framework gives rise to many new partial differential equations on graphs, most notably a new (Laplacian based) wave equation; this wave equation gives rise to a partial improvement on the Chung-Faber-Manteuffel diameter/eigenvalue bound in graph theory, and the Chung-Grigoryan-Yau and (in a certain case) Bobkov-Ledoux distance/eigenvalue bounds in analysis. Our framework also allows most techniques for the non-linear p-Laplacian in analysis to be easily carried over to graph theory.