In this paper, we report the first quantum key-recovery attack on a symmetric
block cipher design, using classical queries only, with a more than quadratic
time speedup compared to the best classical attack.

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We study the 2XOR-Cascade construction of Gav{z}i and Tessaro
(EUROCRYPT~2012). It is a key length extension technique which provides an
n-bit block cipher with 5n/2 bits of security out of an n-bit block cipher with
2n bits of key, with a security proof in the ideal model. We show that the
offline-Simon algorithm of Bonnetain et al. (ASIACRYPT~2019) can be extended
to, 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 assumed
that doubling the key sizes is a sufficient precaution. This is because
Grover’s quantum search algorithm, and its derivatives, can only reach a
quadratic speedup at most. Our attack shows that the structure of some
symmetric constructions can be exploited to overcome this limit. In particular,
the 2XOR-Cascade cannot be used to generically strengthen block ciphers against
quantum adversaries, as it would offer only the same security as the block
cipher itself.

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