Simultaneous approximation problems of p -adic numbers and p -adic knapsack cryptosystems - Alice in p -adic numberland

In this paper we construct the multi-dimensional p -adic approximation lattices by using simultaneous approximation problems (SAP) of p -adic numbers and we estimate the l ∞ norm of the p -adic SAP solutions theoretically by applying Dirichlet’s principle and numerically by using the LLL algorithm. By using the SAP solutions as private keys, the security of which depends on NP-hardness of SAP or the shortest vector problems (SVP) of p -adic lattices, we propose a p -adic knapsack cryptosystem with commitment schemes, in which the sender Alice prepares ciphertexts and the verification keys in her p -adic numberland.