LEDAkem: a post-quantum key encapsulation mechanism based on QC-LDPC codes

This work presents a new code-based key encapsulation mechanism (KEM) called LEDAkem. It is built on the Niederreiter cryptosystem and relies on quasi-cyclic low-density parity-check codes as secret codes, providing high decoding speeds and compact keypairs. LEDAkem uses ephemeral keys to foil known statistical attacks, and takes advantage of a new decoding algorithm that provides faster decoding than the classical bit-flipping decoder commonly adopted in this kind of systems. The main attacks against LEDAkem are investigated, taking into account quantum speedups. Some instances of LEDAkem are designed to achieve different security levels against classical and quantum computers. Some performance figures obtained through an efficient C99 implementation of LEDAkem are provided.Comment: 21 pages, 3 table

Some Notes on Code-Based Cryptography

This thesis presents new cryptanalytic results in several areas of coding-based cryptography. In addition, we also investigate the possibility of using convolutional codes in code-based public-key cryptography. The first algorithm that we present is an information-set decoding algorithm, aiming towards the problem of decoding random linear codes. We apply the generalized birthday technique to information-set decoding, improving the computational complexity over previous approaches. Next, we present a new version of the McEliece public-key cryptosystem based on convolutional codes. The original construction uses Goppa codes, which is an algebraic code family admitting a well-defined code structure. In the two constructions proposed, large parts of randomly generated parity checks are used. By increasing the entropy of the generator matrix, this presumably makes structured attacks more difficult. Following this, we analyze a McEliece variant based on quasi-cylic MDPC codes. We show that when the underlying code construction has an even dimension, the system is susceptible to, what we call, a squaring attack. Our results show that the new squaring attack allows for great complexity improvements over previous attacks on this particular McEliece construction. Then, we introduce two new techniques for finding low-weight polynomial multiples. Firstly, we propose a general technique based on a reduction to the minimum-distance problem in coding, which increases the multiplicity of the low-weight codeword by extending the code. We use this algorithm to break some of the instances used by the TCHo cryptosystem. Secondly, we propose an algorithm for finding weight-4 polynomials. By using the generalized birthday technique in conjunction with increasing the multiplicity of the low-weight polynomial multiple, we obtain a much better complexity than previously known algorithms. Lastly, two new algorithms for the learning parities with noise (LPN) problem are proposed. The first one is a general algorithm, applicable to any instance of LPN. The algorithm performs favorably compared to previously known algorithms, breaking the 80-bit security of the widely used (512,1/8) instance. The second one focuses on LPN instances over a polynomial ring, when the generator polynomial is reducible. Using the algorithm, we break an 80-bit security instance of the Lapin cryptosystem

BIKE: Bit Flipping Key Encapsulation

Submission to the NIST post quantum standardization proces

FuLeeca: A Lee-based Signature Scheme

In this work we introduce a new code-based signature scheme, called \textsf{FuLeeca}, based on the NP-hard problem of finding codewords of given Lee-weight. The scheme follows the Hash-and-Sign approach applied to quasi-cyclic codes. Similar approaches in the Hamming metric have suffered statistical attacks, which revealed the small support of the secret basis. Using the Lee metric, we are able to thwart such attacks. We use existing hardness results on the underlying problem and study adapted statistical attacks. We propose parameters for \textsf{FuLeeca}~and compare them to an extensive list of proposed post-quantum secure signature schemes including the ones already standardized by NIST. This comparison reveals that \textsf{FuLeeca}~is competitive. For example, for NIST category I, i.e., 160 bit of classical security, we obtain an average signature size of 1100 bytes and public key sizes of 1318 bytes. Comparing the total communication cost, i.e., the sum of the signature and public key size, we see that \textsf{FuLeeca} is only outperformed by Falcon while the other standardized schemes Dilithium and SPHINCS+ show larger communication costs than \textsf{FuLeeca}

Faster Constant-Time Decoder for MDPC Codes and Applications to BIKE KEM

BIKE is a code-based key encapsulation mechanism (KEM) that was recently selected as an alternate candidate by the NIST’s standardization process on post-quantum cryptography. This KEM is based on the Niederreiter scheme instantiated with QC-MDPC codes, and it uses the BGF decoder for key decapsulation. We discovered important limitations of BGF that we describe in detail, and then we propose a new decoding algorithm for QC-MDPC codes called PickyFix. Our decoder uses two auxiliary iterations that are significantly different from previous approaches and we show how they can be implemented efficiently. We analyze our decoder with respect to both its error correction capacity and its performance in practice. When compared to BGF, our constant-time implementation of PickyFix achieves speedups of 1.18, 1.29, and 1.47 for the security levels 128, 192 and 256, respectively

Compact McEliece keys based on Quasi-Dyadic Srivastava codes

The McEliece cryptosystem is one of the few systems to be considered secure against Quantum attacks. The original scheme is built upon Goppa codes and produces very large keys, hence latest research has focused mainly on trying to reduce the public key size. Previous proposals tried to replace the class of Goppa codes with other families of codes, but this revealed to be an insecure choice. In this paper we introduce a construction based on Generalized Srivastava codes, a large class which include Goppa codes as a special case, that allows relatively short public keys without being vulnerable to known structural attacks

Curves, codes, and cryptography

This thesis deals with two topics: elliptic-curve cryptography and code-based cryptography. In 2007 elliptic-curve cryptography received a boost from the introduction of a new way of representing elliptic curves. Edwards, generalizing an example from Euler and Gauss, presented an addition law for the curves x2 + y2 = c2(1 + x2y2) over non-binary fields. Edwards showed that every elliptic curve can be expressed in this form as long as the underlying field is algebraically closed. Bernstein and Lange found fast explicit formulas for addition and doubling in coordinates (X : Y : Z) representing (x, y) = (X/Z, Y/Z) on these curves, and showed that these explicit formulas save time in elliptic-curve cryptography. It is easy to see that all of these curves are isomorphic to curves x2 + y2 = 1 + dx2y2 which now are called "Edwards curves" and whose shape covers considerably more elliptic curves over a finite field than x2 + y2 = c2(1 + x2y2). In this thesis the Edwards addition law is generalized to cover all curves ax2 +y2 = 1+dx2y2 which now are called "twisted Edwards curves." The fast explicit formulas for addition and doubling presented here are almost as fast in the general case as they are for the special case a = 1. This generalization brings the speed of the Edwards addition law to every Montgomery curve. Tripling formulas for Edwards curves can be used for double-base scalar multiplication where a multiple of a point is computed using a series of additions, doublings, and triplings. The use of double-base chains for elliptic-curve scalar multiplication for elliptic curves in various shapes is investigated in this thesis. It turns out that not only are Edwards curves among the fastest curve shapes, but also that the speed of doublings on Edwards curves renders double bases obsolete for this curve shape. Elliptic curves in Edwards form and twisted Edwards form can be used to speed up the Elliptic-Curve Method for integer factorization (ECM). We show how to construct elliptic curves in Edwards form and twisted Edwards form with large torsion groups which are used by the EECM-MPFQ implementation of ECM. Code-based cryptography was invented by McEliece in 1978. The McEliece public-key cryptosystem uses as public key a hidden Goppa code over a finite field. Encryption in McEliece’s system is remarkably fast (a matrix-vector multiplication). This system is rarely used in implementations. The main complaint is that the public key is too large. The McEliece cryptosystem recently regained attention with the advent of post-quantum cryptography, a new field in cryptography which deals with public-key systems without (known) vulnerabilities to attacks by quantum computers. The McEliece cryptosystem is one of them. In this thesis we underline the strength of the McEliece cryptosystem by improving attacks against it and by coming up with smaller-key variants. McEliece proposed to use binary Goppa codes. For these codes the most effective attacks rely on information-set decoding. In this thesis we present an attack developed together with Daniel J. Bernstein and Tanja Lange which uses and improves Stern’s idea of collision decoding. This attack is faster by a factor of more than 150 than previous attacks, bringing it within reach of a moderate computer cluster. We were able to extract a plaintext from a ciphertext by decoding 50 errors in a [1024, 524] binary code. The attack should not be interpreted as destroying the McEliece cryptosystem. However, the attack demonstrates that the original parameters were chosen too small. Building on this work the collision-decoding algorithm is generalized in two directions. First, we generalize the improved collision-decoding algorithm for codes over arbitrary fields and give a precise analysis of the running time. We use the analysis to propose parameters for the McEliece cryptosystem with Goppa codes over fields such as F31. Second, collision decoding is generalized to ball-collision decoding in the case of binary linear codes. Ball-collision decoding is asymptotically faster than any previous attack against the McEliece cryptosystem. Another way to strengthen the system is to use codes with a larger error-correction capability. This thesis presents "wild Goppa codes" which contain the classical binary Goppa codes as a special case. We explain how to encrypt and decrypt messages in the McEliece cryptosystem when using wild Goppa codes. The size of the public key can be reduced by using wild Goppa codes over moderate fields which is explained by evaluating the security of the "Wild McEliece" cryptosystem against our generalized collision attack for codes over finite fields. Code-based cryptography not only deals with public-key cryptography: a code-based hash function "FSB"was submitted to NIST’s SHA-3 competition, a competition to establish a new standard for cryptographic hashing. Wagner’s generalized birthday attack is a generic attack which can be used to find collisions in the compression function of FSB. However, applying Wagner’s algorithm is a challenge in storage-restricted environments. The FSBday project showed how to successfully mount the generalized birthday attack on 8 nodes of the Coding and Cryptography Computer Cluster (CCCC) at Technische Universiteit Eindhoven to find collisions in the toy version FSB48 which is contained in the submission to NIST