Secure Knapsack Problem Based on Continued Fraction

Abstract

Merkle–Hellman knapsack cryptosystem is a public key cryptosystem, which entails the use of two keys: public and private, the fisrt one used for encryption, while the second one used for decryption. Unfortunately, it is not secure against cryptosystems attacks,where it is broken by Lenstra, Lenstra, and Lovasz ( LLL Algorithm), Adi Shamir. In this paper, we propose a Knapsack–type public key cryptosystems by using a continued fraction, where the continued fraction is used to reduce the coding of plain text into two numbers, regardless of the length of the plain text. We will show that in this paper the Knapsack cryptosystems is secure against the orthogonal lattice attack(LLL Algorithm). Also, the proposed cryptosystems are secured against some attacks (brute–force attack, some known key–recovery attack, frequency attack and quantum attacks). It shows that the continued Fraction provides short plaintext and ciphertext, in which  the encrypted data volume is noticed to be decreased by 60 precent, and this in turn reduces the delay time.