# Learning Plaintext in Galbraith’s LWE Cryptosystem

### Tikaram Sanyashi, Sreyans Nahata, Rushang Dhanesha, Bernard Menezes

#### Abstract

Unlike many widely used cryptosytems, Learning with Errors (LWE) - based cryptosystems are known to be invulnerable to quantum computers. Galbraith’s Binary LWE (GB-LWE) was proposed to reduce the large key size of the original LWE scheme by over two orders of magnitude. In GB-LWE, recovering the plaintext from the ciphertext involves solving for the binary vector x in the equation xA = b (A, a 640×256 binary matrix and b, a 256 element integer vector are knowns). Previously, lattice-based attacks on binary matrices larger than 400 × 256 were found to be infeasible. Linear programming was proposed and shown to handle significantly larger matrices but its success rate for 640 × 256 matrices was found to be negligible. Our strategy involves identification of regimes L, M and H within the output (based on LP relaxation) where the mis-prediction rates are low, medium or high respectively. Bits in the output vector are guessed and removed to create and solve a reduced instance. We report extensive experimental results on prediction accuracy and success probability as a function of number of bits removed in L, M and H. We identify trade-offs between lower execution time and greater probability of success. Our success probability is much higher than previous efforts and its execution time of 1 day with 150 cores is a partial response to the challenge posed in (Galbraith, 2013) to solve a random 640 × 256 instance using “current computing facilities in less than a year”.

Download#### Paper Citation

#### in Harvard Style

Sanyashi T., Dhanesha R. and Menezes B. (2018). **Learning Plaintext in Galbraith’s LWE Cryptosystem**.In *Proceedings of the 15th International Joint Conference on e-Business and Telecommunications - Volume 1: SECRYPT,* ISBN 978-989-758-319-3, pages 559-565. DOI: 10.5220/0006909405590565

#### in Bibtex Style

@conference{secrypt18,

author={Tikaram Sanyashi and Rushang Dhanesha and Bernard Menezes},

title={Learning Plaintext in Galbraith’s LWE Cryptosystem},

booktitle={Proceedings of the 15th International Joint Conference on e-Business and Telecommunications - Volume 1: SECRYPT,},

year={2018},

pages={559-565},

publisher={SciTePress},

organization={INSTICC},

doi={10.5220/0006909405590565},

isbn={978-989-758-319-3},

}

#### in EndNote Style

TY - CONF

JO - Proceedings of the 15th International Joint Conference on e-Business and Telecommunications - Volume 1: SECRYPT,

TI - Learning Plaintext in Galbraith’s LWE Cryptosystem

SN - 978-989-758-319-3

AU - Sanyashi T.

AU - Dhanesha R.

AU - Menezes B.

PY - 2018

SP - 559

EP - 565

DO - 10.5220/0006909405590565