MDS matrix

Abstract: Twofish is a 128-bit block cipher that accepts a variable-length key up to 256 bits. The cipher is a 16-round Feistel network with a bijective F function made up of four key-dependent 8-by-8-bit S-boxes, a fixed 4-by-4 maximum distance separable matrix over GF(2), a pseudo-Hadamard transform, bitwise rotations, and a carefully designed key schedule. A fully optimized implementation of Twofish encrypts on a Pentium Pro at 17.8 clock cycles per byte, and an 8-bit smart card implementation encrypts at 1660 clock cycles per byte. Twofish can be implemented in hardware in 14000 gates. The design of both the round function and the key schedule permits a wide variety of tradeoffs between speed, software size, key setup time, gate count, and memory. We have extensively cryptanalyzed Twofish; our best attack breaks 5 rounds with 2 chosen plaintexts and 2 effort.

Abstract: We present the new block cipher SHARK. This cipher combines highly non-linear substitution boxes and maximum distance separable error correcting codes (MDS-codes) to guarantee a good diffusion. The cipher is resistant against differential and linear cryptanalysis after a small number of rounds. The structure of SHARK is such that a fast software implementation is possible, both for the encryption and the decryption. Our C-implementation of SHARK runs more than four times faster than SAFER and IDEA on a 64-bit architecture.

Abstract: Although linear perfect diffusion primitives, i.e. MDS matrices, are widely used in block ciphers, e.g. AES, very little systematic work has been done on how to find “efficient” ones. In this paper we attempt to do so by considering software implementations on various platforms. These considerations lead to interesting combinatorial problems: how to maximize the number of occurrences of 1 in those matrices, and how to minimize the number of pairwise different entries. We investigate these problems and construct efficient 4 × 4 and 8 × 8 MDS matrices to be used e.g. in block ciphers.

Abstract: Many modern block ciphers use maximum distance separable (MDS) matrices as the main part of their diffusion layers In this paper, we propose a new class of diffusion layers constructed from several rounds of Feistel-like structures whose round functions are linear We investigate the requirements of the underlying linear functions to achieve the maximal branch number for the proposed 4×4 words diffusion layer The proposed diffusion layers only require word-level XORs, rotations, and they have simple inverses They can be replaced in the diffusion layer of the block ciphers MMB and Hierocrypt to increase their security and performance, respectively Finally, we try to extend our results for up to 8×8 words diffusion layers

Abstract: Diffusion layers with maximum branch numbers are widely used in block ciphers and hash functions. In this paper, we construct recursive diffusion layers using Linear Feedback Shift Registers (LFSRs). Unlike the MDS matrix used in AES, whose elements are limited in a finite field, a diffusion layer in this paper is a square matrix composed of linear transformations over a vector space. Perfect diffusion layers with branch numbers from 5 to 9 are constructed. On the one hand, we revisit the design strategy of PHOTON lightweight hash family and the work of FSE 2012, in which perfect diffusion layers are constructed by one bundle-based LFSR. We get better results and they can be used to replace those of PHOTON to gain smaller hardware implementations. On the other hand, we investigate new strategies to construct perfect diffusion layers using more than one bundle-based LFSRs. Finally, we construct perfect diffusion layers by increasing the number of iterations and using bit-level LFSRs. Since most of our proposals have lightweight examples corresponding to 4-bit and 8-bit Sboxes, we expect that they will be useful in designing (lightweight) block ciphers and (lightweight) hash functions.