Abstract

Knapsack-based cryptosystems had been viewed as the most attractive and the most promising asymmetric cryptographic algorithms for a long time due to their NP-completeness nature and high speed in encryption/decryption. Unfortunately, most of them are broken for the low-density feature of the underlying knapsack problems. In this paper, we investigate a new easy compact knapsack problem and propose a novel knapsack-based probabilistic public-key cryptosystem in which the cipher-text is non-linear with the plaintext. For properly chosen parameters, the underlying knapsack problem enjoys a high density larger than 1.06 in the worst case. Hence, it is secure against the low-density subset-sum attacks. Our scheme can also defeat other potential attacks such as the brute force attacks and the simultaneous Diophantine approximation attack. Compared with previous knapsack-based cryptosystems, our scheme is efficient and practical.

Keywords: Public-key cryptography; Knapsack cryptosystem; Compact knapsack problem; Low-density subset-sum attack; Simultaneous Diophantine approximation; Lattice basis reduction

Article Outline

1. Introduction

2. Underlying hard problems

2.1. Knapsack problem

2.2. A new easy compact knapsack problem

3. The proposed knapsack-based cryptosystem

3.1. Key generation

3.2. Encryption

3.3. Decryption

3.4. An example

4. Performance and parameters

4.1. Parameter specification

4.2. Performance

5. Security analysis

5.1. Brute-force attacks

5.2. Low-density attacks

5.3. Simultaneous Diophantine approximation attacks

5.3.1. Known m attack

5.3.2. Lagarias’ attack

5.4. Security remarks

6. Comparisons

7. Conclusions

Acknowledgements

Appendix A

A.1. Lattices

A.2. Low-density subset-sum attacks

A.3. Simultaneous Diophantine approximation

References