# American Institute of Mathematical Sciences

August  2018, 12(3): 515-524. doi: 10.3934/amc.2018030

## A note on some algebraic trapdoors for block ciphers

 Department of Informatics, University of Bergen, Norway

Received  June 2017 Revised  March 2018 Published  July 2018

We provide sufficient conditions to guarantee that a translation based cipher is not vulnerable with respect to the partition-based trapdoor. This trapdoor has been introduced, recently, by Bannier et al. (2016) and it generalizes that introduced by Paterson in 1999. Moreover, we discuss the fact that studying the group generated by the round functions of a block cipher may not be sufficient to guarantee security against these trapdoors for the cipher.

Citation: Marco Calderini. A note on some algebraic trapdoors for block ciphers. Advances in Mathematics of Communications, 2018, 12 (3) : 515-524. doi: 10.3934/amc.2018030
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##### References:
AES state
 $V_1$ $V_2$ $V_3$ $V_4$ $V_5$ $V_6$ $V_7$ $V_8$ $V_9$ $V_{10}$ $V_{11}$ $V_{12}$ $V_{13}$ $V_{14}$ $V_{15}$ $V_{16}$
 $V_1$ $V_2$ $V_3$ $V_4$ $V_5$ $V_6$ $V_7$ $V_8$ $V_9$ $V_{10}$ $V_{11}$ $V_{12}$ $V_{13}$ $V_{14}$ $V_{15}$ $V_{16}$
AES wall
 $\color{orange}{V_1}$ $V_2$ $V_3$ $V_4$ $\color{orange}{V_1}$ $V_2$ $V_3$ $V_4$ $\color{orange}{V_1}$ $V_2$ $V_3$ $V_4$ $V_5$ $\color{orange}{V_6}$ $V_7$ $V_8$ $\mathop {SR}\limits_ \mapsto$ $\color{orange}{V_5}$ $V_6$ $V_7$ $V_8$ $\mathop {MC}\limits_ \mapsto$ $\color{orange}{V_5}$ $V_6$ $V_7$ $V_8$ $V_9$ $V_{10}$ $\color{orange}{V_{11}}$ $V_{12}$ $\color{orange}{V_9}$ $V_{10}$ $V_{11}$ $V_{12}$ $\color{orange}{V_9}$ $V_{10}$ $V_{11}$ $V_{12}$ $V_{13}$ $V_{14}$ $V_{15}$ $\color{orange}{V_{16}}$ $\color{orange}{V_{13}}$ $V_{14}$ $V_{15}$ $V_{16}$ $\color{orange}{V_{13}}$ $V_{14}$ $V_{15}$ $V_{16}$
 $\color{orange}{V_1}$ $V_2$ $V_3$ $V_4$ $\color{orange}{V_1}$ $V_2$ $V_3$ $V_4$ $\color{orange}{V_1}$ $V_2$ $V_3$ $V_4$ $V_5$ $\color{orange}{V_6}$ $V_7$ $V_8$ $\mathop {SR}\limits_ \mapsto$ $\color{orange}{V_5}$ $V_6$ $V_7$ $V_8$ $\mathop {MC}\limits_ \mapsto$ $\color{orange}{V_5}$ $V_6$ $V_7$ $V_8$ $V_9$ $V_{10}$ $\color{orange}{V_{11}}$ $V_{12}$ $\color{orange}{V_9}$ $V_{10}$ $V_{11}$ $V_{12}$ $\color{orange}{V_9}$ $V_{10}$ $V_{11}$ $V_{12}$ $V_{13}$ $V_{14}$ $V_{15}$ $\color{orange}{V_{16}}$ $\color{orange}{V_{13}}$ $V_{14}$ $V_{15}$ $V_{16}$ $\color{orange}{V_{13}}$ $V_{14}$ $V_{15}$ $V_{16}$
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