Lightweight Multiplication in $$GF(2^n)$$ with Applications to MDS Matrices

Beierle, Christof; Kranz, Thorsten; Leander, Gregor

doi:10.1007/978-3-662-53018-4_23

Cited by 46 publications

(67 citation statements)

References 22 publications

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“…We do not discuss efficiency issues but the theory accumulated and discussed here should provide an idea towards efficiency. The readers may look at [6] and the references mentioned therein for efficient implementation of MDS matrices. We revisit the results discussed in [2] and find a gap in one of the lemmas in that paper.…”

Section: Resultsmentioning

confidence: 99%

Cryptographically significant mds matrices over finite fields: A brief survey and some generalized results

Gupta¹,

Pandey²,

Ray³

et al. 2019

Advances in Mathematics of Communications

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A matrix is MDS or super-regular if and only if every square submatrices of it are nonsingular. MDS matrices provide perfect diffusion in block ciphers and hash functions. In this paper we provide a brief survey on cryptographically significant MDS matrices-a first to the best of our knowledge. In addition to providing a summary of existing results, we make several contributions. We exhibit some deep and nontrivial interconnections between different constructions of MDS matrices. For example, we prove that all known Vandermonde constructions are basically equivalent to Cauchy constructions. We prove some folklore results which are used in MDS matrix literature. Wherever possible, we provide some simpler alternative proofs. We do not discuss efficiency issues or hardware implementations; however, the theory accumulated and discussed here should provide an easy guide towards efficient implementations.

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Section: Resultsmentioning

confidence: 99%

Cryptographically significant mds matrices over finite fields: A brief survey and some generalized results

Gupta¹,

Pandey²,

Ray³

et al. 2019

Advances in Mathematics of Communications

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“…In this case, x ⊕ x = 0 and obviously wt(x , x) ≥ wt(x ⊕ x ). Thus, considering equation 3, we have (5) wt…”

Section: Cryptographic Properties Of L Nmentioning

confidence: 99%

“…Equations (4) and (5) implies that b g ≥ b f . In the case of b f = w f + 1, as w f = w g , we have b g = b f , by Lemma 3.3.…”

Section: Cryptographic Properties Of L Nmentioning

confidence: 99%

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Cryptographic properties of cyclic binary matrices

Rishakani¹,

Dehnavi²,

Shamsabad³

et al. 2021

Advances in Mathematics of Communications

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Many modern symmetric ciphers apply MDS or almost MDS matrices as diffusion layers. The performance of a diffusion layer depends on its diffusion property measured by branch number and implementation cost which is usually measured by the number of XORs required. As the implementation cost of MDS matrices of large dimensions is high, some symmetric ciphers use binary matrices as diffusion layers to trade-off efficiency versus diffusion property. In the current paper, we investigate cyclic binary matrices (CBMs for short), mathematically. Based upon this theorical study, we provide efficient matrices with provable lower bound on branch number and minimal number of fixed-points. We consider the product of sparse CBMs to construct efficiently implementable matrices with the desired cryptographic properties.

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“…With lightweight cryptography in mind, there has been a substantial amount of publications in the last few years on constructions for lightweight MDS mappings, see e.g., [BKL16;GPP11;LS16;LW16]. This has led to a better understanding of the implementation cost of such mappings in relation to their dimensions: the number and size of elements (e.g., 4 bytes in MixColumns).…”

Section: Introductionmentioning

confidence: 99%

Column Parity Mixers

Stoffelen

Daemen

2018

ToSC

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We present column parity mixers (CPM), a generalization of the Θ mixing layer that is used in Keccak. Thanks to our description using matrix arithmetic, we can easily derive algebraic, diffusion, and mask propagation properties, leading to a surprising distinction between two types of CPMs. We compare CPMs to other popular types of mixing layers and argue that CPMs can be more efficient. While Keccak has a bit-oriented structure, we make the case that CPMs are also suitable for nibble- or byte-oriented designs. We outline a general substitution-permutation-network-based design strategy using a CPM, for which we show how one can attain strong bounds for differential and linear trails. We apply this strategy concretely to design a 256-bit permutation with an efficient inverse and strong trail bounds. Our permutation design uses a number of ideas that are of independent interest and allows a fast bitsliced implementation that compares quite well with other established ciphers and permutations.

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Lightweight Multiplication in $$GF(2^n)$$ with Applications to MDS Matrices

Cited by 46 publications

References 22 publications

Cryptographically significant mds matrices over finite fields: A brief survey and some generalized results

Cryptographically significant mds matrices over finite fields: A brief survey and some generalized results

Cryptographic properties of cyclic binary matrices

Column Parity Mixers

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