Geodesic
Cryptanalytic invertibility of three-argument functions
Prikladnaya Diskretnaya Matematika. Supplement, no. 17 (2024), pp. 44-48.

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Tests of cryptanalytic invertibility for functions $g:D_1\times D_2\times D_3\to D$ are proposed: 1) function $g$ is invertible with respect to the variable $x_1$ of the type $\forall\forall\exists$ iff there is a mapping $\phi:D_1\times D_2\to D_3$ such that the following condition is satisfied: $$\forall a,c\in D_1 \forall b,d\in D_2 \big(a\neq c\ \Rightarrow\ g(a,b,\phi(a,b))\neq g(c,d,\phi(c,d))\big);$$ 2) function $g$ is invertible with respect to the variable $x_1$ of the type $\forall\exists\forall$ iff there is a mapping $\phi:D_1\to D_2$ such that the following condition is satisfied: $$\forall a,c\in D_1 \forall b,d\in D_3 \big(a\neq c\ \Rightarrow\ g(a,\phi(a),b)\neq g(c,\phi(c),d)\big);$$ 3) function $g$ is invertible with respect to the variable $x_3$ of the type $\forall\exists\forall$ iff there is a mapping $\phi:D_1\to D_2$ such that the following condition is satisfied: $$\forall a,c\in D_1 \forall b,d\in D_3 \big(b\neq d\ \Rightarrow\ g(a,\phi(a),b)\neq g(c,\phi(c),d)\big);$$ 4) function $g$ is invertible with respect to the variable $x_2$ of the type $\exists\forall\forall$ iff there is $a\in D_1$ such that the following condition is satisfied: $$\forall b,d\in D_2 \forall y,z\in D_3 \big(b\neq d\ \Rightarrow\ g(a,b,y)\neq g(a,d,z)\big);$$ 5) function $g$ is invertible with respect to the variable $x_2$ of the type $\exists\forall\exists$ iff there are $a\in D_1$ and a mapping $\phi:D_2\to D_3$ such that the following condition is satisfied: $$\forall b,d\in D_2 \big(b\neq d\ \Rightarrow\ g(a,b,\phi(b))\neq g(a,d,\phi(d))\big).$$ Algorithms for constructing a recovering function and generating invertible functions are formulated too.
Keywords: cryptanalytic invertibility, invertibility test, recovering function. 
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     author = {I. A. Pankratova and A. D. Sorokoumova},
     title = {Cryptanalytic invertibility of three-argument functions},
     journal = {Prikladnaya Diskretnaya Matematika. Supplement},
     pages = {44--48},
     publisher = {mathdoc},
     number = {17},
     year = {2024},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/PDMA_2024_17_a10/}
}
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I. A. Pankratova; A. D. Sorokoumova. Cryptanalytic invertibility of three-argument functions. Prikladnaya Diskretnaya Matematika. Supplement, no. 17 (2024), pp. 44-48. http://geodesic.mathdoc.fr/item/PDMA_2024_17_a10/

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