Code-based Cryptosystems Research Articles

Connecting the Complexity of MQ- and Code-Based Cryptosystems

Abstract We study the connection between the MQ problem and the de­coding problem, through the intermediate MRHS representation. The main goal of this study is to explicitly bound the complexity of solving MQ systems with decoding tools. The main observation is that although the MQ problem over GF(2) can be efficiently transformed to syndrome decoding, the existing general decoding methods are not suitable to solve the system as efficiently as expected from the MQ representation.

Punctured Reed–Muller code‐based McEliece cryptosystems

The authors propose new McEliece cryptosystems based on punctured Reed-Muller (RM) codes. They successfully show that the commonly known attacks, such as the Minder-Shokrollahi attack, the Chizhov-Borodin attack, and the square code attack, are ineffective against the proposed RM code-based McEliece cryptosystem. We developed an optimal puncturing scheme to prevent the above-mentioned attacks for the proposed RM code-based cryptosystems in a sense that the exact locations of puncturing positions with the minimum number of punctured columns of the generator matrix should be found for attacking. It is important to carry out the minimum number of punctures, however, as code modification resulting from puncturing can reduce security. Additionally, the square code attack can be prevented in the proposed RM code-based McEliece cryptosystems by using both the proposed puncturing method and random insertion methods.

Code‐based security with random interleaving

In this paper, a new symmetric key cryptosystem based on linear codes is presented. Unlike most code-based cryptosystems, the code is known publicly. The encryption is based on randomly interleaving two codewords containing random errors. This is the first code-based cryptosystem based on random interleaving. The key size and security of the proposed scheme are analysed, and it is shown that allowing the code to be public results in a significantly smaller key than other code-based cryptosystems. Furthermore, it is proven that the best possible chosen plaintext attack does not affect the security.

On lower bounds for information set decoding over 𝔽<SUB align="right">q and on the effect of partial knowledge

Code-based cryptosystems are promising candidates for post-quantum cryptography since they are fast, require only basic arithmetic because their security is well understood. The increasing number of cryptographic schemes based on codes over fields other than 𝔽2 presents, however, security issues that are not relevant in the case of binary codes; the security of such constructions, therefore, requires separate assessment. Information set decoding (ISD) is one of the most important generic attacks against code-based cryptosystems. We give lower bounds for ISD over 𝔽q, thereby anticipating future software and hardware improvements. Our results allow to compute conservative parameters for cryptographic applications. While most security proofs assume that an attacker does not have any additional information about the secret, we show that in certain scenarios an attacker can gain partial knowledge of the secret. We present how this knowledge can be used to improve the efficiency of an attack and give new bounds for the complexity of such an attack. In this paper, we analyse two types of partial knowledge including concrete scenarios and give an idea how to prevent the leakage of such knowledge to an attacker.

An Efficient Public Key Encryption Scheme Based on QC-MDPC Lattices

In this paper, we introduce a new family of lattices, namely QC-MDPC lattices, which are a special case of LDPC lattices, and an improved bit flipping algorithm for decoding of these lattices. Encoding and decoding implementations of QC-MDPC lattices are practical in high dimensions. Indeed, to take advantage of practical decoding, we use “Construction-A” lattices which makes a tight connection between the structure of lattices and codes. Using these features, we design a lattice-based public key encryption scheme enjoying linear encryption and decryption complexities. The proposed scheme has a reasonable key size due to the sparseness of the parity-check matrix, and the quasi-cyclic structure of the parity-check and generator matrices. Besides, the message expansion of the proposed scheme is smaller than other lattice-based and code-based cryptosystems with comparative parameters. All these features provide a lattice-based public key encryption scheme with reasonable key size, linear encryption, and decryption algorithms and small message expansion. On the other hand, we show that the cryptosystem is resistant against all known attacks both on lattice-based and code-based cryptosystems for different levels of security.

Public-key code-based cryptography

The possibility of a quantum computer threatens the existence of many asymmetric cryptographic primitives. An important advantage of code cryptosystems is the high resistance to the quantum cryptanalysis. Therefore it is in great demand increasing the speed of the code based primitives and their protection from classical cryptanalysis. Code-Based Public-Key Cryptosystems based on algebraic coding are considered in this paper. In addition, the current state, the existing contradictions and prospects of practical use for the post-quantum period are studied. We consider estimation of resistance to attack permutation decoding and computational complexity compared to the RSA scheme. We proposed a new code cryptosystem, it significantly increases the relative information performance and keeps the main advantages of resistance to classical and quantum cryptanalysis.

On lower bounds for information set decoding over 𝔽<SUB align="right">q and on the effect of partial knowledge

Code-based cryptosystems are promising candidates for post-quantum cryptography since they are fast, require only basic arithmetic because their security is well understood. The increasing number of cryptographic schemes based on codes over fields other than F

Joint channel coding‐cryptography based on random insertions and deletions in quasi‐cyclic‐low‐density parity check codes

In this paper, a new secure channel coding scheme is presented which randomly inserts and deletes bits in a codeword of a quasi-cyclic-low-density parity check (QC-LDPC) code. It is shown that the key size is smaller than other code-based cryptosystems based on permutation and scrambling matrices. The positions of the inserted and deleted bits are determined using a secret key. It is shown that the error performance of the resulting code after the insertions and deletions is better than a random low-density parity check code with similar parameters. An important advantage of this cryptosystem is that even if the QC-LDPC code is revealed, the system remains secure. Furthermore, the proposed approach using insertions and deletions can be employed with other classes of error correcting codes.

Implementing QC-MDPC McEliece Encryption

With respect to performance, asymmetric code-based cryptography based on binary Goppa codes has been reported as a highly interesting alternative to RSA and ECC. A major drawback is still the large keys in the range between 50 and 100KB that prevented real-world applications of code-based cryptosystems so far. A recent proposal by Misoczki et al. showed that quasi-cyclic moderate-density parity-check (QC-MDPC) codes can be used in McEliece encryption, reducing the public key to just 0.6KB to achieve an 80-bit security level. In this article, we provide optimized decoding techniques for MDPC codes and survey several efficient implementations of the QC-MDPC McEliece cryptosystem. This includes high-speed and lightweight architectures for reconfigurable hardware, efficient coding styles for ARM’s Cortex-M4 microcontroller, and novel high-performance software implementations that fully employ vector instructions. Finally, we conclude that McEliece encryption in combination with QC-MDPC codes not only enables high-performance implementations but also allows for lightweight designs on a wide range of different platforms.

Critical attacks in code-based cryptography

Code-based cryptographic schemes are promising candidates for post-quantum cryptography since they are fast, require only basic arithmetic, and have a well understood security. While there is strong evidence that cryptosystems like McEliece and Niederreiter are secure, they have certain weaknesses when used without semantic conversions. Critical attacks generally can't be avoided by increasing the key size of several code-based cryptosystems. In this paper we present a survey on critical attacks in code-based cryptography and we propose a specific conversion with a smaller redundancy of data than Korara's et al. and which protects against CCA2. Our purpose is to evaluate three cryptosystems: McEliece, Niederreiter and HyMES. We analyse their security against several models such as: Broadcast, Known Partial plaintext, Message-resend, Related-message, Chosen ciphertext, Lunchtime, Reaction attack and Malleability. Our work follows a first work done by Imai and Kobara 2001 which does not cover the whole known attacks and does not deal with the HyMES scheme.

New secure channel coding scheme based on randomly punctured quasi‐cyclic‐low density parity check codes

A new joint cryptography-channel coding technique is introduced which employs punctured quasi-cyclic-low density parity check (QC-LDPC) codes obtained from extended difference families. The absence of permutation and scrambling matrices reduces the key size compared with similar code-based cryptosystems, while having an acceptable level of security. The main advantage of this system is that, provided the system parameters are chosen appropriately, even if the code employed is revealed the system remains secure. Performance results are presented which show that the punctured code outperforms a random low density parity check (LDPC) code of the same length and rate.

Scaling efficient code-based cryptosystems for embedded platforms

We describe a family of efficient codes for cryptographic purposes and dedicated algorithms for their manipulation. Our proposal is especially tailored for highly constrained platforms, becoming competitive with conventional schemes.

Selecting parameters for secure McEliece-based cryptosystems

In 1994, Shor showed that quantum computers will be able to break cryptosystems based on the problems of integer factorization and the discrete logarithm, for example, RSA or ECC. Code-based cryptosystems are promising alternatives to public-key schemes built on these problems, and they are believed to be secure against quantum computer attacks. In this paper, we solve the problem of selecting optimal parameters for the McEliece cryptosystem that are expected to provide security at least until a given year and give detailed recommendations. Our analysis is based on the lower bound complexity estimates by Sendrier and Finiasz, and the security requirements model proposed by Lenstra and Verheul. This security model uses assumptions about Moore’s Law and other developments in order to estimate the attained security level for a given year.