Twisted Edwards curves over finite fields have attracted great interest for their efficient and unified addition formula. In this paper, we consider twisted Edwards curves over local fields and introduce a cryptosystem based on quotient groups of twisted Edwards curves over local fields. From the study of formal groups of twisted Edwards curves and twisted Edwards curves over local fields, we give the choice of cryptographic groups.An element in these groups can be uniformly represented by two n digit p-adic numbers, whereas an element in the elliptic curves in Weierstrass form over local fields is represented by a 3n-2 digit p-adic number and a4n-3 digit p-adic number. In the cryptography on elliptic curves in Weierstrass form over local fields, five cases for different input point pairs in computing points addition have to be considered and sometimes points have to be lifted. In the cryptography on twisted Edwards curves over local fields, the addition formula is simple,unified, and complete, which is efficient, does not need lifting points, and is against the side channel analysis.Finally, a speedy point multiplication algorithm and some concrete instances are given. Twisted Edwards curves over finite fields have attracted great interest for their efficient and unified addition formula. In this paper, we consider twisted Edwards curves over local fields and introduce a cryptosystem based on quotient groups of twisted Edwards curves over local fields. From the study of formal groups of twisted Edwards curves and twisted Edwards curves over local fields, we give the choice of cryptographic groups. An element in these groups can be represented by two n digit p-adic numbers, an an element in the elliptic curves in Weierstrass form over local fields is represented by a 3n-2 digit p-adic number and a4n-3 digit p-adic number. In the cryptography on elliptic curves in Weierstrass form over local fields, five cases for different input point pairs in computing points addition additionally have to be considered and sometimes points have to be lifted. In the cryptography on twisted Edwards curves over local fields, the addition formula is simple, unified, and complete, which is efficient, does not need lifting points, and is against the side channel analysis. Finally, a speedy point multiplication algorithm and some concrete instances are given.