Equivalence Algorithms: Difference between revisions
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==The to and from algorithm== | ==The to and from algorithm== | ||
This algorithm is | This algorithm is presented at eurocrypt 2003 <ref name="back">Biryukov, Alex, et al. "A toolbox for cryptanalysis: Linear and affine equivalence algorithms." Advances in Cryptology—EUROCRYPT 2003: International Conference on the Theory and Applications of Cryptographic Techniques, Warsaw, Poland, May 4–8, 2003 Proceedings 22. Springer Berlin Heidelberg, 2003. </ref>. This algorithm is mainly intended for when the boolean functions are permutation, and we will start by assuming <math>f </math> and <math>g </math> are permutations. | ||
The idea of the algorithm is to go use information gather about <math>A_1 </math> to deduce information about <math>A_2 </math> and the other way around. To see how this can work let's say we know some values of <math>A_1 </math>, lets say <math>A_1(x_1)=y_1, A_2(x_2) = y_2,..., A_1(x_n) = y_n </math>. | |||
Revision as of 14:10, 19 November 2024
The hierarchy of equivalences
Given two vectorial boolean functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f,g : F_2^n \rightarrow F_2^n } there are various ways to define equivalence between Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g } . We will study the algorithms for determining Linear, Affine, Extended Affine and CCZ equivalence between vectorial boolean functions.
Linear Equivalence
Given two vectorial boolean functions Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f } and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle g } we want to determine if there exist Linear permutations Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1} and Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_2 } such that Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f = A_2 \circ g \circ A_1 } .
The to and from algorithm
This algorithm is presented at eurocrypt 2003 [1]. This algorithm is mainly intended for when the boolean functions are permutation, and we will start by assuming Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle f } and are permutations.
The idea of the algorithm is to go use information gather about Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1 } to deduce information about Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_2 } and the other way around. To see how this can work let's say we know some values of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1 } , lets say Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle A_1(x_1)=y_1, A_2(x_2) = y_2,..., A_1(x_n) = y_n } .
- ↑ Biryukov, Alex, et al. "A toolbox for cryptanalysis: Linear and affine equivalence algorithms." Advances in Cryptology—EUROCRYPT 2003: International Conference on the Theory and Applications of Cryptographic Techniques, Warsaw, Poland, May 4–8, 2003 Proceedings 22. Springer Berlin Heidelberg, 2003.