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In this paper, we report the first quantum key-recovery attack on a symmetricblock cipher design, using classical queries only, with a more than quadratictime speedup compared to the best classical attack.  We study the 2XOR-Cascade construction of Ga\v{z}i and Tessaro(EUROCRYPT~2012). It is a key length extension technique which provides ann-bit block cipher with 5n/2 bits of security out of an n-bit block cipher with2n bits of key, with a security proof in the ideal model. We show that theoffline-Simon algorithm of Bonnetain et al. (ASIACRYPT~2019) can be extendedto, in particular, attack this construction in quantum time \~O($2^n$),providing a 2.5 quantum speedup over the best classical attack.  Regarding post-quantum security of symmetric ciphers, it is commonly assumedthat doubling the key sizes is a sufficient precaution. This is becauseGrover's quantum search algorithm, and its derivatives, can only reach aquadratic speedup at most. Our attack shows that the structure of somesymmetric constructions can be exploited to overcome this limit. In particular,the 2XOR-Cascade cannot be used to generically strengthen block ciphers againstquantum adversaries, as it would offer only the same security as the blockcipher itself.

Beyond quadratic speedups in quantum attacks on symmetric schemes