SHORTEST VECTOR PROBLEMS OF p-ADIC RANDOM LATTICES AND THEIR APPLICATION TO A p-ADIC KNAPSACK TYPE CRYPTOSYSTEM

We construct the multi-dimensional p-adic approximation lattices by using the simultaneous approximation problems of random p-adic numbers given by a p-adic logistic map, which is well known as a chaos generator. For the knapsack type matrices given by these random lattices we numerically investigate their security against the low density attacks, using LLL algorithms, where the density is the ratio of the bit size of a plaintext to the bit size of its cyphertext. Using a p-adic decreasing sequence instead of a super increasing sequence of usual integers, we propose a p-adic compact knapsack cryptosystem, which is secure against Shamir's attacks. Applying our numerical results, we can estimate the secure values of parameters in our cryptosystem against the low density attacks.