An interesting challenge for the cryptography community is to design authentication protocols that are so simple that a human can execute them without relying on a fully trusted computer. We propose several candidate authentication protocols for a setting in which the human user can only receive assistance from a semi-trusted computer---a computer that stores information and performs computations correctly but does not provide confidentiality. Our schemes use a semi-trusted computer to store and display public challenges $C_i\in[n]^k$. The human user memorizes a random secret mapping $\sigma:[n]\rightarrow \mathbb{Z}_d$ and authenticates by computing responses $f(\sigma(C_i))$ to a sequence of public challenges where $f:\mathbb{Z}_d^k\rightarrow \mathbb{Z}_d$ is a function that is easy for the human to evaluate. We prove that any statistical adversary needs to sample $m=\tilde{\Omega}(n^{s(f)})$ challenge-response pairs to recover $\sigma$, for a security parameter $s(f)$ that depends on two key properties of $f$. Our lower bound generalizes recent results of Feldman et al. who proved analogous results for the special case $d=2$. To obtain our results, we apply the general hypercontractivity theorem to lower bound the statistical dimension of the distribution over challenge-response pairs induced by $f$ and $\sigma$. Our statistical dimension lower bounds apply to arbitrary functions $f:\mathbb{Z}_d^k\rightarrow \mathbb{Z}_d$ (not just to functions that are easy for a human to evaluate). As an application, we propose a family of human computable password functions $f_{k_1,k_2}$ in which the user needs to perform $2k_1+2k_2+1$ primitive operations (e.g., adding two digits or looking up a secret value $\sigma(i)$), and we show that $s(f) = \min\{k_1+1, (k_2+1)/2\}$. For these schemes, we prove that forging passwords is equivalent to recovering the secret mapping. Thus, our human computable password schemes can maintain strong security guarantees even after an adversary has observed the user login to many different accounts.

*This is joint work with Manuel Blum, Anupam Datta and Santosh Vempala*