  Quantum computers, that may become available one day, would impact many
scientific fields, most notably cryptography since many asymmetric primitives
are insecure against an adversary with quantum capabilities. Cryptographers are
already anticipating this threat by proposing and studying a number of
potentially quantum-safe alternatives for those primitives. On the other hand,
symmetric primitives seem less vulnerable against quantum computing: the main
known applicable result is Grover's algorithm that gives a quadratic speed-up
for exhaustive search.
  In this work, we examine more closely the security of symmetric ciphers
against quantum attacks. Since our trust in symmetric ciphers relies mostly on
their ability to resist cryptanalysis techniques, we investigate quantum
cryptanalysis techniques. More specifically, we consider quantum versions of
differential and linear cryptanalysis. We show that it is usually possible to
use quantum computations to obtain a quadratic speed-up for these attack
techniques, but the situation must be nuanced: we don't get a quadratic
speed-up for all variants of the attacks. This allows us to demonstrate the
following non-intuitive result: the best attack in the classical world does not
necessarily lead to the best quantum one. We give some examples of application
on ciphers LAC and KLEIN. We also discuss the important difference between an
adversary that can only perform quantum computations, and an adversary that can
also make quantum queries to a keyed primitive.
