Quantum Secure Non-Malleable Codes in the Split-State Model

Non-malleable codes introduced by Dziembowski et al. (2018) encode a classical message
$S$
in a manner such that the tampered codeword either decodes to the original message
$S$
or a message that is unrelated/independent of
$S$
. Constructing non-malleable codes for various tampering function families has received significant attention in the recent years. We consider the well studied (2-part) split-state model, in which the message
$S$
is encoded into two parts
$X$
and
$Y$
, and the adversary is allowed to arbitrarily tamper with each
$X$
and
$Y$
individually. Non-malleable codes in the split-state model have found applications in other important security notions like non-malleable commitments and non-malleable secret sharing. Thus, it is vital to understand if such non-malleable codes are secure against quantum adversaries. We consider the security of non-malleable codes in the split-state model when the adversary is allowed to make use of arbitrary entanglement to tamper the parts
$X$
and
$Y$
. We construct explicit quantum secure non-malleable codes in the split-state model. Our construction of quantum secure non-malleable codes is based on the recent construction of quantum secure 2-source non-malleable extractors by Boddu et al. (2021). 1) We extend the connection of Cheraghchi and Guruswami (2016) between 2-source non-malleable extractors and non-malleable codes in the split-state model in the classical setting to the quantum setting, i.e. we show that explicit quantum secure 2-source non-malleable extractors in
$(k_{1},k_{2})- \mathsf {qpa- state} $
framework of Boddu et al. (2021) give rise to explicit quantum secure non-malleable codes in the split-state model. 2) We construct the first quantum secure non-malleable code with efficient encoding and decoding procedures for message length
$m=n^{\Omega (1)}$
, error
$\varepsilon =2^{-n^{\Omega (1)}}$
and codeword of size
$2n$
. Prior to this work, it remained open to provide such quantum secure non-malleable code even for a single bit message in the split-state model. 3) We also study its natural extension when the tampering of the codeword is performed
$t$
-times. We construct quantum secure one-many non-malleable code with efficient encoding and decoding procedures for
$t=n^{\Omega (1)}$
, message length
$m=n^{\Omega (1)}$
, error
$\varepsilon =2^{-n^{\Omega (1)}}$
and codeword of size
$2n$
. 4) As an application, we also construct the first quantum secure 2-out-of-2 non-malleable secret sharing scheme for message/secret length
$m=n^{\Omega (1)}$
, error
$\varepsilon =2^{-n^{\Omega (1)}}$
and share of size
$n$
.