A Note on 5-bit Quadratic Permutations’ Classification

Božilov, Dušan; Bilgin, Begül; Sahin, Hacı Ali

doi:10.46586/tosc.v2017.i1.398-404

Cited by 8 publications

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“…Similarly, applying the first approach over 4 × 4 S-boxes to obtain 5 × 5 but restricted only to the affine and quadratic S-boxes gives the results shown in Table 3. We obtain 23 out of the 75 quadratic classes given in [1]. In addition, it is clear from this construction why for the classes Q 5 30 and Q 5 32 no uniform sharing with 3 shares was found in [1].…”

Section: Application Of Shannon's Expansion To S-boxesmentioning

confidence: 91%

“…Recall that a uniform sharing with 3 shares exists for Q 3 1 , Q 3 2 , but not for Q 3 3 ; and a uniform sharing with 4, 5 and more shares exists for all 3 of them. Also recall that a uniform sharing with 3 shares exists for Q 4 4 , Q [1]. Moreover, all 5-bit quadratic permutation classes have a uniform sharing with 4 and more shares.…”

Section: Threshold Implementations and Uniform Sharingmentioning

confidence: 94%

“…Here, t i represents Boolean functions of n inputs and F is written instead of F + 1 for simplicity. Note that compared to the construction (1) used in [3] to get from 3 × 3 an 4 × 4 S-box and similarly in [1] from 4 × 4 an 5 × 5 S-box, the construction (5) extends it to allow F to be any Boolean function on n variables. Now we will show how the constructions (3) or ( 5) can be used to find uniform sharings.…”

Section: Increasing the Size Of An S-box By Onementioning

confidence: 99%

“…There is a transformation [3] which expands the 3-bit classes Q 3 1 , Q 3 2 , and Q 3 3 into Q 4 4 , Q 4 12 and Q 4 300 correspondingly. That is, given a 3-bit permutation S(x 1 , x 2 , x 3 ) = (y 1 , y 2 , y 3 ), its 4-bit extension is generated by S(x 1 , x 2 , x 3 , x 4 ) = (y 1 , y 2 , y 3 , x 4 ).Recently a classification of all quadratic 5 × 5 bijective S-boxes was presented in [1]. The authors have also pointed out that the 5-bit classes Q 5 1 , Q 5 3 , Q 5 4 , Q 5 7 , Q 5 13 and Q 5 30 are extensions of the 4-bit quadratic classes Q 4 4 , Q 4 294 , Q 4 12 , Q 4 299 , Q 4 293and Q 4 300 from [3] respectively.…”

mentioning

confidence: 99%

“…Recently a classification of all quadratic 5 × 5 bijective S-boxes was presented in [1]. The authors have also pointed out that the 5-bit classes Q 5 1 , Q 5 3 , Q 5 4 , Q 5 7 , Q 5 13 and Q 5 30 are extensions of the 4-bit quadratic classes Q 4 4 , Q 4 294 , Q 4 12 , Q 4 299 , Q 4 293…”

mentioning

confidence: 99%

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Constructions of S-boxes with uniform sharing

et al. 2019

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In this paper we focus on S-box constructions. We consider the uniformity property of an S-box which plays an important role in Threshold Implementations (TI). Most papers so far have studied TI sharings for given S-boxes. We proceed in the opposite way: starting from n-bit S-boxes with known sharings we construct new (n + 1)-bit S-boxes from them with the desired sharings. In addition, we investigate the self-equivalency of S-boxes and show some interesting properties.maximal algebraic degree of a balanced n-variable Boolean function is n − 1 [5,8]. De Cannière lists 302 equivalence classes for the 4 × 4 bijections: the class of affine functions, 6 classes containing quadratic functions and the remaining 295 classes containing cubic functions. There is a transformation [3] which expands the 3-bit classes Q 3 1 , Q 3 2 , and Q 3 3 into Q 4 4 , Q 4 12 and Q 4 300 correspondingly. That is, given a 3-bit permutation S(x 1 , x 2 , x 3 ) = (y 1 , y 2 , y 3 ), its 4-bit extension is generated by S(x 1 , x 2 , x 3 , x 4 ) = (y 1 , y 2 , y 3 , x 4 ).Recently a classification of all quadratic 5 × 5 bijective S-boxes was presented in [1]. The authors have also pointed out that the 5-bit classes Q 5 1 , Q 5 3 , Q 5 4 , Q 5 7 , Q 5 13 and Q 5 30 are extensions of the 4-bit quadratic classes Q 4 4 , Q 4 294 , Q 4 12 , Q 4 299 , Q 4 293and Q 4 300 from [3] respectively. That is, given a 4-bit permutation S(x 1 , x 2 , x 3 , x 4 ) = (y 1 , y 2 , y 3 , y 4 ), its 5-bit extension is generated by S(x 1 , x 2 , x 3 , x 4 , x 5 ) = (y 1 , y 2 , y 3 , y 4 , x 5 ). Letx = (x 1 , ..., x n ) then the method used in the above mentioned publications can be summarized as follows S(x, x n+1 ) = S 1 (x) for the first n bits (1) = x n+1 for the (n + 1)-st bitAnother well known construction is the so-called Shannon expansion.

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Section: Application Of Shannon's Expansion To S-boxesmentioning

confidence: 91%

Section: Threshold Implementations and Uniform Sharingmentioning

confidence: 94%

Section: Increasing the Size Of An S-box By Onementioning

confidence: 99%

mentioning

confidence: 99%

mentioning

confidence: 99%

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Constructions of S-boxes with uniform sharing

et al. 2019

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Constructing TI-Friendly Substitution Boxes Using Shift-Invariant Permutations

Gao

Oswald

2019

Topics in Cryptology – CT-RSA 2019

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The threat posed by side channels requires ciphers that can be efficiently protected in both software and hardware against such attacks. In this paper, we proposed a novel Sbox construction based on iterations of shift-invariant quadratic permutations and linear diffusions. Owing to the selected quadratic permutations, all of our Sboxes enable uniform 3-share threshold implementations, which provide first order SCA protections without any fresh randomness. More importantly, because of the "shift-invariant" property, there are ample implementation trade-offs available, in software as well as hardware. We provide implementation results (software and hardware) for a four-bit and an eight-bit Sbox, which confirm that our constructions are competitive and can be easily adapted to various platforms as claimed. We have successfully verified their resistance to first order attacks based on real acquisitions. Because there are very few studies focusing on software-based threshold implementations, our software implementations might be of independent interest in this regard.

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Changing of the Guards: A Simple and Efficient Method for Achieving Uniformity in Threshold Sharing

Daemen

2017

Lecture Notes in Computer Science

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A Note on 5-bit Quadratic Permutations’ Classification

Cited by 8 publications

References 0 publications

Constructions of S-boxes with uniform sharing

Constructions of S-boxes with uniform sharing

Constructing TI-Friendly Substitution Boxes Using Shift-Invariant Permutations

Changing of the Guards: A Simple and Efficient Method for Achieving Uniformity in Threshold Sharing

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