Paper 2023/1689

Revisiting the Boomerang Attack from a Perspective of 3-differential

Libo Wang, University of Hyogo, Kobe, Japan
Ling Song, Jinan University, Guangzhou, China
Baofeng Wu, Institute of Information Engineering, CAS, Beijing, China; School of Cyber Security, UCAS, Beijing, China
Mostafizar Rahman, University of Hyogo, Kobe, Japan
Takanori Isobe, University of Hyogo, Kobe, Japan; NICT, Tokyo, Japan

In this paper, inspired by the work of Beyne and Rijmen at CRYPTO 2022, we explore the accurate probability of $d$-differential in the fixed-key model. The theoretical foundations of our method are based on a special matrix $-$ quasi-$d$-differential transition matrix, which is a natural extension of the quasidifferential transition matrix. The role of quasi-$d$-differential transition matrices in polytopic cryptananlysis is analogous to that of correlation matrices in linear cryptanalysis. Therefore, the fixed-key probability of a $d$-differential can be exactly expressed as the sum of the correlations of its quasi-$d$-differential trails. Then we revisit the boomerang attack from a perspective of 3-differential. Different from previous works, the probability of a boomerang distinguisher can be exactly expressed as the sum of the correlations of its quasi-$3$-differential trails without any assumptions in our work. In order to illustrate our theory, we apply it to the lightweight block cipher GIFT. It is interesting to find the probability of every optimal 3-differential characteristic of an existing 2-round boomerang is zero, which can be seen as an evidence that the security of block ciphers adopting half-round key XOR might be overestimated previously to some extent in differential-like attacks.

Available format(s)
Secret-key cryptography
Publication info
Published elsewhere. IEEE Transactions on Information Theory
Boomerang attack$d$-differentialHypothesis of stochastic equivalenceCorrelation matrixQuasidifferential transition matrix{\sf GIFT}
Contact author(s)
wanglibo12b @ gmail com
songling qs @ gmail com
wubaofeng @ iie ac cn
mrahman454 @ gmail com
takanori isobe @ ai u-hyogo ac jp
2023-11-03: approved
2023-11-01: received
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      author = {Libo Wang and Ling Song and Baofeng Wu and Mostafizar Rahman and Takanori Isobe},
      title = {Revisiting the Boomerang Attack from a Perspective of 3-differential},
      howpublished = {Cryptology ePrint Archive, Paper 2023/1689},
      year = {2023},
      note = {\url{}},
      url = {}
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