Exhaustive Search for Small Dimension Recursive MDS Diffusion Layers for Block Ciphers and Hash Functions

This article presents a new algorithm to find MDS matrices that are well suited for use as a diffusion layer in lightweight block ciphers. Using an recursive construction, it is possible to obtain matrices with a very compact description. Classical field multiplications can also be replaced by simple F2-linear transformations (combinations of XORs and shifts) which are much lighter. Using this algorithm, it was possible to design a 16x16 matrix on a 5-bit alphabet, yielding an efficient 80-bit diffusion layer with maximal branch number.Comment: Published at ISIT 201

Direct Construction of Recursive MDS Diffusion Layers using Shortened BCH Codes

MDS matrices allow to build optimal linear diffusion layers in block ciphers. However, MDS matrices cannot be sparse and usually have a large description, inducing costly software/hardware implementations. Recursive MDS matrices allow to solve this problem by focusing on MDS matrices that can be computed as a power of a simple companion matrix, thus having a compact description suitable even for constrained environ- ments. However, up to now, finding recursive MDS matrices required to perform an exhaustive search on families of companion matrices, thus limiting the size of MDS matrices one could look for. In this article we propose a new direct construction based on shortened BCH codes, al- lowing to efficiently construct such matrices for whatever parameters. Unfortunately, not all recursive MDS matrices can be obtained from BCH codes, and our algorithm is not always guaranteed to find the best matrices for a given set of parameters.Comment: Best paper award; Carlos Cid and Christian Rechberger. 21st International Workshop on Fast Software Encryption, FSE 2014, Mar 2014, London, United Kingdom. springe

The Design Space of Lightweight Cryptography

International audienceFor constrained devices, standard cryptographic algorithms can be too big, too slow or too energy-consuming. The area of lightweight cryptography studies new algorithms to overcome these problems. In this paper, we will focus on symmetric-key encryption, authentication and hashing. Instead of providing a full overview of this area of research, we will highlight three interesting topics. Firstly, we will explore the generic security of lightweight constructions. In particular, we will discuss considerations for key, block and tag sizes, and explore the topic of instantiating a pseudorandom permutation (PRP) with a non-ideal block cipher construction. This is inspired by the increasing prevalence of lightweight designs that are not secure against related-key attacks, such as PRINCE, PRIDE or Chaskey. Secondly, we explore the efficiency of cryptographic primitives. In particular, we investigate the impact on efficiency when the input size of a primitive doubles. Lastly, we provide some considerations for cryptographic design. We observe that applications do not always use cryptographic algorithms as they were intended, which negatively impacts the security and/or efficiency of the resulting implementations

Lightweight Design Choices for LED-like Block Ciphers

Serial matrices are a preferred choice for building diffusion layers of lightweight block ciphers as one just needs to implement the last row of such a matrix. In this work we analyze a new class of serial matrices which are the lightest possible $4 \times 4$ serial matrix that can be used to build diffusion layers. With this new matrix we show that block ciphers like LED can be implemented with a reduced area in hardware designs, though it has to be cycled for more iterations. Further, we suggest the usage of an alternative S-box to the standard S-box used in LED with similar cryptographic robustness, albeit having lesser area footprint. Finally, we combine these ideas in an end-end FPGA based prototype of LED. We show that with these optimizations, there is a reduction of $16$ in area footprint of one round implementation of LED

Design of Lightweight Linear Diffusion Layers from Near-MDS Matrices

Near-MDS matrices provide better trade-offs between security and efficiency compared to constructions based on MDS matrices, which are favored for hardwareoriented designs. We present new designs of lightweight linear diffusion layers by constructing lightweight near-MDS matrices. Firstly generic n×n near-MDS circulant matrices are found for 5 ≤ n ≤9. Secondly, the implementation cost of instantiations of the generic near-MDS matrices is examined. Surprisingly, for n = 7, 8, it turns out that some proposed near-MDS circulant matrices of order n have the lowest XOR count among all near-MDS matrices of the same order. Further, for n = 5, 6, we present near-MDS matrices of order n having the lowest XOR count as well. The proposed matrices, together with previous construction of order less than five, lead to solutions of n×n near-MDS matrices with the lowest XOR count over finite fields F2m for 2 ≤ n ≤ 8 and 4 ≤ m ≤ 2048. Moreover, we present some involutory near-MDS matrices of order 8 constructed from Hadamard matrices. Lastly, the security of the proposed linear layers is studied by calculating lower bounds on the number of active S-boxes. It is shown that our linear layers with a well-chosen nonlinear layer can provide sufficient security against differential and linear cryptanalysis

Feistel Like Construction of Involutory Binary Matrices With High Branch Number

In this paper, we propose a generic method to construct involutory binary matrices from a three round Feistel scheme with a linear round function. We prove bounds on the maximum achievable branch number (BN) and the number of fixed points of our construction. We also define two families of efficiently implementable round functions to be used in our method. The usage of these families in the proposed method produces matrices achieving the proven bounds on branch numbers and fixed points. Moreover, we show that BN of the transpose matrix is the same with the original matrix for the function families we defined. Some of the generated matrices are \emph{Maximum Distance Binary Linear} (MDBL), i.e. matrices with the highest achievable BN. The number of fixed points of the generated matrices are close to the expected value for a random involution. Generated matrices are especially suitable for utilising in bitslice block ciphers and hash functions. They can be implemented efficiently in many platforms, from low cost CPUs to dedicated hardware

Efficient MDS Diffusion Layers Through Decomposition of Matrices

Diffusion layers are critical components of symmetric ciphers. MDS matrices are diffusion layers of maximal branch number which have been used in various symmetric ciphers. In this article, we examine decomposition of cyclic matrices from mathematical viewpoint and based on that, we present new cyclic MDS matrices. From the aspect of implementation, the proposed matrices have lower implementation costs both in software and hardware, compared to what is presented in cryptographic literature, up to our knowledge

On the Construction of Near-MDS Matrices

The optimal branch number of MDS matrices makes them a preferred choice for designing diffusion layers in many block ciphers and hash functions. However, in lightweight cryptography, Near-MDS (NMDS) matrices with sub-optimal branch numbers offer a better balance between security and efficiency as a diffusion layer, compared to MDS matrices. In this paper, we study NMDS matrices, exploring their construction in both recursive and nonrecursive settings. We provide several theoretical results and explore the hardware efficiency of the construction of NMDS matrices. Additionally, we make comparisons between the results of NMDS and MDS matrices whenever possible. For the recursive approach, we study the DLS matrices and provide some theoretical results on their use. Some of the results are used to restrict the search space of the DLS matrices. We also show that over a field of characteristic 2, any sparse matrix of order $n\geq 4$ with fixed XOR value of 1 cannot be an NMDS when raised to a power of $k\leq n$. Following that, we use the generalized DLS (GDLS) matrices to provide some lightweight recursive NMDS matrices of several orders that perform better than the existing matrices in terms of hardware cost or the number of iterations. For the nonrecursive construction of NMDS matrices, we study various structures, such as circulant and left-circulant matrices, and their generalizations: Toeplitz and Hankel matrices. In addition, we prove that Toeplitz matrices of order $n>4$ cannot be simultaneously NMDS and involutory over a field of characteristic 2. Finally, we use GDLS matrices to provide some lightweight NMDS matrices that can be computed in one clock cycle. The proposed nonrecursive NMDS matrices of orders 4, 5, 6, 7, and 8 can be implemented with 24, 50, 65, 96, and 108 XORs over $\mathbb{F}_{2^4}$, respectively

On the Direct Construction of MDS and Near-MDS Matrices

The optimal branch number of MDS matrices makes them a preferred choice for designing diffusion layers in many block ciphers and hash functions. Consequently, various methods have been proposed for designing MDS matrices, including search and direct methods. While exhaustive search is suitable for small order MDS matrices, direct constructions are preferred for larger orders due to the vast search space involved. In the literature, there has been extensive research on the direct construction of MDS matrices using both recursive and nonrecursive methods. On the other hand, in lightweight cryptography, Near-MDS (NMDS) matrices with sub-optimal branch numbers offer a better balance between security and efficiency as a diffusion layer compared to MDS matrices. However, no direct construction method is available in the literature for constructing recursive NMDS matrices. This paper introduces some direct constructions of NMDS matrices in both nonrecursive and recursive settings. Additionally, it presents some direct constructions of nonrecursive MDS matrices from the generalized Vandermonde matrices. We propose a method for constructing involutory MDS and NMDS matrices using generalized Vandermonde matrices. Furthermore, we prove some folklore results that are used in the literature related to the NMDS code

Block Ciphers - Focus On The Linear Layer (feat. PRIDE): Full Version

The linear layer is a core component in any substitution-permutation network block cipher. Its design significantly influences both the security and the efficiency of the resulting block cipher. Surprisingly, not many general constructions are known that allow to choose trade-offs between security and efficiency. Especially, when compared to Sboxes, it seems that the linear layer is crucially understudied. In this paper, we propose a general methodology to construct good, sometimes optimal, linear layers allowing for a large variety of trade-offs. We give several instances of our construction and on top underline its value by presenting a new block cipher. PRIDE is optimized for 8-bit micro-controllers and significantly outperforms all academic solutions both in terms of code size and cycle count