## Cryptology ePrint Archive: Report 2019/520

A Note on Sub-Gaussian Random Variables

Benjamin M. Case and Colin Gallagher and Shuhong Gao

Abstract: A sub-Gaussian distribution is any probability distribution that has tails bounded by a Gaussian and has a mean of zero. It is well known that the sum of independent sub-Gaussians is again sub-Gaussian. This note generalizes this result to sums of sub- Gaussians that may not be independent, under the assumption a certain conditional distribution is also sub-Gaussian. This general result is useful in the study of noise growth in (fully) homomorphic encryption schemes [CGHX19, CGGI17], and hopefully useful for other applications.

Category / Keywords: foundations / sub-Gaussians, fully homomorphic encryption FHE, boostrapping, error analysis, lattices, TFHE