Difference between revisions of "Boomerang uniformity"

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(Created page with "=Background and definitions= The Boomerang attack, introduced in 1999 by Wagner <ref name="wagnerBoomerangAttack>Wagner D. The boomerang attack.In Lars R. Knudsen, editor, FSE...")
 
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** For <math>F'</math> an affine equivalent permutation, <math>F'=A_2\circ F\circ A_1</math>, we have <math>T_{F'}(a,b)=T_F(L_1(a),L_2^{-1}(b))</math>, with <math>L_i</math> the linear part of <math>A_i</math>.
 
** For <math>F'</math> an affine equivalent permutation, <math>F'=A_2\circ F\circ A_1</math>, we have <math>T_{F'}(a,b)=T_F(L_1(a),L_2^{-1}(b))</math>, with <math>L_i</math> the linear part of <math>A_i</math>.
 
** For the inverse we have <math>T_{F^{-1}}(a,b)=T_F(b,a)</math>.
 
** For the inverse we have <math>T_{F^{-1}}(a,b)=T_F(b,a)</math>.
* <math>\delta_F\le\beta_F</math> and <math>\delta_F=2</math> if and only if <math>\beta_F=2</math>.
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* Relation with the differential uniformity: <math>\delta_F\le\beta_F</math> and <math>\delta_F=2</math> if and only if <math>\beta_F=2</math>.
 
* <math>T_F(a,b)=|\{ (x,y) : F(x+a)+F(y+a)=b,F(x)+F(y)=b \}|</math>.
 
* <math>T_F(a,b)=|\{ (x,y) : F(x+a)+F(y+a)=b,F(x)+F(y)=b \}|</math>.
 
* If <math>F</math> is a power permutation, then <math>\beta_F=\max_{b\neq0}T(1,b)</math>.
 
* If <math>F</math> is a power permutation, then <math>\beta_F=\max_{b\neq0}T(1,b)</math>.
 
* If <math>F</math> is a quadratic permutation, then <math>\delta_F\le\beta_F\le\delta_F(\delta_F-1)</math>.
 
* If <math>F</math> is a quadratic permutation, then <math>\delta_F\le\beta_F\le\delta_F(\delta_F-1)</math>.

Latest revision as of 09:46, 23 September 2019

Background and definitions

The Boomerang attack, introduced in 1999 by Wagner [1], is a cryptanalysis technique against block ciphers based on differential cryptanalysis. To study the resistance to this attack, Cid et al.[2] introduced the Boomerang Connectivity Table (BCT). Next, Boura and Canteaut[3] , introduced the notion of boomerang uniformity.

For a permutation , the Boomerang Connectivity Table (BCT) is given by a table ,

.

The boomerang uniformity of is the maximal value, i.e.

Main properties

For a permutation, the following properties on the boomerang uniformity were proven.

  • The boomerang uniformity is invariant for inverse and affine equivalence but not for EA- and CCZ-equivalence.
    • For an affine equivalent permutation, , we have , with the linear part of .
    • For the inverse we have .
  • Relation with the differential uniformity: and if and only if .
  • .
  • If is a power permutation, then .
  • If is a quadratic permutation, then .
  • Wagner D. The boomerang attack.In Lars R. Knudsen, editor, FSE'99, vol. 1636 of LNCS, pp. 156-170. Springer, Heidelberg, March 1999
  • Cid C., Huang T., Peyrin T., Sasaki Y., Song L. Boomerang connectivity table: A new cryptanalysis tool. EUROCRYPT 2018, Part II, vol. 10821 of LNCS, pp. 683-714. Springer, Heidelberg, 2018
  • Boura C., Canteaut A. On the boomerang uniformity of cryptographic Sboxes. IACR Transaction on Symmetric Cryptology, pp. 290-310, Sep 2018