Geodesic
On the anisometric index of a~transformation
Prikladnaya Diskretnaya Matematika. Supplement, no. 10 (2017), pp. 25-27.

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Many papers deal with finding distances between a transformation and an affine or imprimitive group. In cryptography, these results are often connected with investigations of linear and homomorphic models of block ciphers. Besides, to provide adequate resistance of block ciphers to generalizations of linear and homomorphic attacks, good cryptographic transformations must diffuse the structures associated with the affine and imprimitive groups. Some structures of block ciphers can be linked up with an isometry group of a discrete metric space, but in cryptography, such structures are seldom considered. In this paper, for a transformation $g\colon V_n(2)\to V_n(2)$ and a partition $\mathbf W$ of the set $(V_n(2))^2$ of the metric space $(\mu,V_n(2))$, we introduce a measure that characterizes the diffusion degree of $\mathbf W$ in relation to $g$. The measure is called the anisometric index of the transformation $g$. We get upper bounds of the anisometric index for some classes of transformations. Further, we show that the anisometric index can be expressed in terms of elements of the difference distribution table. We also get relations between anisometric indexes of affine-equivalent transformations. In addition, we investigate links between two classes of permutations. The first class consists of all permutations that have the largest Hamming distance from imprimitive groups $S_{2^{n - 1}}\wr S_2$, $S_2\wr S_{2^{n - 1}}$. The second class consists of all permutations that have the largest anisometric index. In particular, we show that, for some metrics, these classes are the same ones.
Keywords: Hamming distance, isometry group, difference distribution table
Mots-clés : imprimitive group. 
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B. A. Pogorelov; M. A. Pudovkina. On the anisometric index of a~transformation. Prikladnaya Diskretnaya Matematika. Supplement, no. 10 (2017), pp. 25-27. http://geodesic.mathdoc.fr/item/PDMA_2017_10_a8/

[1] Pogorelov B. A., Pudovkina M. A., “O rasstoyaniyakh ot podstanovok do imprimitivnykh grupp pri fiksirovannoi sisteme imprimitivnosti”, Diskretnaya matematika, 25:3 (2013), 78–95 | DOI | MR

[2] Pogorelov B. A., Osnovy teorii grupp podstanovok. Ch. 1. Obschie voprosy, V/ch 33965, M., 1986, 316 pp.

[3] Pogorelov B. A., Pudovkina M. A., “Naturalnye metriki i ikh svoistva. Ch. 2. Metriki tipa Khemminga”, Matematicheskie voprosy kriptografii, 3:1 (2012), 71–95

[4] Pogorelov B. A., “Podmetriki metriki Khemminga i teorema A. A. Markova”, Trudy po diskretnoi matematike, 9, 2006, 190–219

[5] Pogorelov B. A., Pudovkina M. A., “Podmetriki metriki Khemminga i preobrazovaniya, rasprostranyayuschie iskazheniya v zadannoe chislo raz”, Trudy po diskretnoi matematike, 10, 2007, 202–238