Lightweight Diffusion Layer: Importance of Toeplitz Matrices

Sarkar, Sumanta; Syed, Habeeb

doi:10.46586/tosc.v2016.i1.95-113

Cited by 33 publications

(18 citation statements)

References 12 publications

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“…In this section, first, we focus on generating

4 \times 4

involutory MDS matrices over

F_{2^{4}}

and

F_{2^{8}}

in GHadamard matrix form with the minimum value of XOR count. We present

4 \times 4

involutory MDS matrices over

F_{2^{4}}

, which meet the least XOR count 64 (

= 16 + 4 \times 3 \times 4

) as reported in [11] and provide an example with XOR count 158 (

= 62 + 4 \times 3 \times 8

) for a

4 \times 4

involutory MDS matrix over

F_{2^{8}}

. Then, we apply our proposed matrix form to generate

8 \times 8

involutory and non‐involutory MDS matrices over

F_{2^{4}}

with the minimum value of XOR count.…”

Section: Experimental Results For Lightweight Mds Matricesmentioning

confidence: 99%

“…These attempts were mainly based on search and the two outstanding studies [11, 14] focused on finding lightweight (involutory) MDS matrices. In [11], Sarkar and Syed provided a theoretical construction of a class of involutory MDS matrices and gave two examples of

4 \times 4

involutory and MDS matrices over

F_{2^{4}}

and

F_{2^{8}}

with lower XOR counts (they actually improved the best‐known result of XOR count for

4 \times 4

involutory matrix over

F_{2^{4}}

). In Examples 1 and 2, we show that the involutory and MDS matrices given in [11] are also generated by

4 \times 4

GHadamard matrix form.…”

Section: New Matrix Form Ghadamard Matrixmentioning

confidence: 99%

“…The lightest MDS circulant matrices in view of the requiring exclusive OR (XOR) gates were given over

F_{2^{4}}

and

F_{2^{8}}

. In [11, 12], Toeplitz matrices with a theoretical point of view to construct MDS matrices were studied and shown that implementation properties of those matrices are comparable with the previous studies.…”

Section: Introductionmentioning

confidence: 99%

“…Then, the equality also holds for a GHadamard matrix

G H

, i.e.

false(G H false)^{2} = c^{2} I

. (e) Using GHadamard matrix form, we generate

4 \times 4

involutory MDS matrices over

F_{2^{4}}

with the best‐known result of XOR count 64 [11]. (f) Using GHadamard matrix form, we generate a

4 \times 4

involutory MDS matrix over

F_{2^{8}}

with XOR count 158. (g) Using GHadamard matrix form, we improve the best‐known result of XOR count for

8 \times 8

involutory MDS matrices over

F_{2^{4}}

from 512 [14] to 407. (h) Using GHadamard matrix form, we improve the best‐known result of XOR count for

8 \times 8

non‐involutory MDS matrices over

F_{2^{4}}

from 432 [14] to 380.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Generalisation of Hadamard matrix to generate involutory MDS matrices for lightweight cryptography

Pehlıvanoğlu¹,

Sakallı²,

Akleylek³

et al. 2018

IET Information Security

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“…In this section, first, we focus on generating

4 \times 4

involutory MDS matrices over

F_{2^{4}}

and

F_{2^{8}}

in GHadamard matrix form with the minimum value of XOR count. We present

4 \times 4

involutory MDS matrices over

F_{2^{4}}

, which meet the least XOR count 64 (

= 16 + 4 \times 3 \times 4

) as reported in [11] and provide an example with XOR count 158 (

= 62 + 4 \times 3 \times 8

) for a

4 \times 4

involutory MDS matrix over

F_{2^{8}}

. Then, we apply our proposed matrix form to generate

8 \times 8

involutory and non‐involutory MDS matrices over

F_{2^{4}}

with the minimum value of XOR count.…”

Section: Experimental Results For Lightweight Mds Matricesmentioning

confidence: 99%

4 \times 4

involutory and MDS matrices over

F_{2^{4}}

and

F_{2^{8}}

with lower XOR counts (they actually improved the best‐known result of XOR count for

4 \times 4

involutory matrix over

F_{2^{4}}

). In Examples 1 and 2, we show that the involutory and MDS matrices given in [11] are also generated by

4 \times 4

GHadamard matrix form.…”

Section: New Matrix Form Ghadamard Matrixmentioning

confidence: 99%

“…The lightest MDS circulant matrices in view of the requiring exclusive OR (XOR) gates were given over

F_{2^{4}}

and

F_{2^{8}}

Section: Introductionmentioning

confidence: 99%

“…Then, the equality also holds for a GHadamard matrix

G H

, i.e.

false(G H false)^{2} = c^{2} I

. (e) Using GHadamard matrix form, we generate

4 \times 4

involutory MDS matrices over

F_{2^{4}}

with the best‐known result of XOR count 64 [11]. (f) Using GHadamard matrix form, we generate a

4 \times 4

involutory MDS matrix over

F_{2^{8}}

with XOR count 158. (g) Using GHadamard matrix form, we improve the best‐known result of XOR count for

8 \times 8

involutory MDS matrices over

F_{2^{4}}

from 512 [14] to 407. (h) Using GHadamard matrix form, we improve the best‐known result of XOR count for

8 \times 8

non‐involutory MDS matrices over

F_{2^{4}}

from 432 [14] to 380.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Generalisation of Hadamard matrix to generate involutory MDS matrices for lightweight cryptography

Pehlıvanoğlu¹,

Sakallı²,

Akleylek³

et al. 2018

IET Information Security

View full text Add to dashboard Cite

“…In addition to construction methods described for MDS matrices, recently, MDS construction methods have evolved to find MDS matrices with minimal XOR counts [21], which is a metric used in the estimation of hardware implementation cost. In the literature, some studies focusing on generating MDS matrices with low/minimum XOR counts are given in [20,[22][23][24]. In this paper, we focus on a complementary method to generate isomorphic MDS matrices from existing ones to be applied to any MDS matrix generated by any construction method, which makes it generic over all construction methods.…”

Section: Introductionmentioning

confidence: 99%

On the automorphisms and isomorphisms of MDS matrices and their efficient implementations

Sakallı¹,

Akleylek²,

Akkanat³

et al. 2020

Turk J Elec Eng & Comp Sci

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In this paper, we explicitly define the automorphisms of MDS matrices over the same binary extension field.By extending this idea, we present the isomorphisms between MDS matrices over F2m and MDS matrices over F 2 mt , where t ≥ 1 and m > 1 , which preserves the software implementation properties in view of XOR operations and table lookups of any given MDS matrix over F2m . Then we propose a novel method to obtain distinct functions related to these automorphisms and isomorphisms to be used in generating isomorphic MDS matrices (new MDS matrices in view of implementation properties) using the existing ones. The comparison with the MDS matrices used in AES, ANUBIS, and subfield-Hadamard construction shows that we generate an involutory 4 × 4 MDS matrix over F 2 8 (from an involutory 4 × 4 MDS matrix over F 2 4 ) whose required number of XOR operations is the same as that of ANUBIS and the subfield-Hadamard construction, and better than that of AES. The proposed method, due to its ground field structure, is intended to be a complementary method for the current construction methods in the literature.

show abstract

Direct Constructions of (Involutory) MDS Matrices from Block Vandermonde and Cauchy-Like Matrices

Liu

2018

Arithmetic of Finite Fields

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Lightweight Diffusion Layer: Importance of Toeplitz Matrices

Cited by 33 publications

References 12 publications

Generalisation of Hadamard matrix to generate involutory MDS matrices for lightweight cryptography

Generalisation of Hadamard matrix to generate involutory MDS matrices for lightweight cryptography

On the automorphisms and isomorphisms of MDS matrices and their efficient implementations

Direct Constructions of (Involutory) MDS Matrices from Block Vandermonde and Cauchy-Like Matrices

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