TY - GEN

T1 - General Impossibility of Group Homomorphic Encryption in the Quantum World

AU - Armknecht, Frederik

AU - Gagliardoni, Tommaso

AU - Katzenbeisser, Stefan

AU - Peter, Andreas

N1 - eemcs-eprint-24762

PY - 2014

Y1 - 2014

N2 - Group homomorphic encryption represents one of the most important building blocks in modern cryptography. It forms the basis of widely-used, more sophisticated primitives, such as CCA2-secure encryption or secure multiparty computation. Unfortunately, recent advances in quantum computation show that many of the existing schemes completely break down once quantum computers reach maturity (mainly due to Shor’s algorithm). This leads to the challenge of constructing quantum-resistant group homomorphic cryptosystems.
In this work, we prove the general impossibility of (abelian) group homomorphic encryption in the presence of quantum adversaries, when assuming the IND-CPA security notion as the minimal security requirement. To this end, we prove a new result on the probability of sampling generating sets of finite (sub-)groups if sampling is done with respect to an arbitrary, unknown distribution. Finally, we provide a sufficient condition on homomorphic encryption schemes for our quantum attack to work and discuss its satisfiability in non-group homomorphic cases. The impact of our results on recent fully homomorphic encryption schemes poses itself as an open question.

AB - Group homomorphic encryption represents one of the most important building blocks in modern cryptography. It forms the basis of widely-used, more sophisticated primitives, such as CCA2-secure encryption or secure multiparty computation. Unfortunately, recent advances in quantum computation show that many of the existing schemes completely break down once quantum computers reach maturity (mainly due to Shor’s algorithm). This leads to the challenge of constructing quantum-resistant group homomorphic cryptosystems.
In this work, we prove the general impossibility of (abelian) group homomorphic encryption in the presence of quantum adversaries, when assuming the IND-CPA security notion as the minimal security requirement. To this end, we prove a new result on the probability of sampling generating sets of finite (sub-)groups if sampling is done with respect to an arbitrary, unknown distribution. Finally, we provide a sufficient condition on homomorphic encryption schemes for our quantum attack to work and discuss its satisfiability in non-group homomorphic cases. The impact of our results on recent fully homomorphic encryption schemes poses itself as an open question.

KW - METIS-304104

KW - SCS-Cybersecurity

KW - IR-91147

KW - EWI-24762

U2 - 10.1007/978-3-642-54631-0_32

DO - 10.1007/978-3-642-54631-0_32

M3 - Conference contribution

SN - 978-3-642-54630-3

T3 - Lecture Notes in Computer Science

SP - 556

EP - 573

BT - 17th International Conference on Practice and Theory in Public-Key Cryptography. PKC 2014

PB - Springer

CY - Berlin

Y2 - 26 March 2014 through 28 March 2014

ER -