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Knapsack Public Key Cryptosystems and Diophantine Approximation [chapter]

J. C. Lagarias
1984 Advances in Cryptology  
This paper presents and analyzes cryptanalytic attacks on knapsack public key cryptosystems that are based on ideas from Diophantine approximation.  ...  Shamir's attack on the basic Merkle-Hellman knapsack cryptosystem is shown to depend on the existence of "unusually good" simultaneous Diophantine approximations to a vector constructed from the public  ...  Introduction In 1978 Merkle and Hellman [14] proposed a public key cryptosystem based on the idea of a ''trapdoor'' knapsack.  ... 
doi:10.1007/978-1-4684-4730-9_1 fatcat:mcbyfnxigza4tplfm3gcekxdpy

Page 4939 of Mathematical Reviews Vol. , Issue 86j [page]

1986 Mathematical Reviews  
Author summary: “We present and analyze cryptanalytic attacks on knapsack public key cryptosystems that are based on ideas from Diophantine approximation. A.  ...  These probabilities are analyzed and shown to be negligibly small.” Lagarias, J. C. (1-BELL) 86j:94050 Knapsack public key cryptosystems and Diophantine approximation (extended abstract).  ... 

Cryptanalysis of a Public-key Cryptosystem Using Lattice Basis Reduction Algorithm [article]

Roohallah Rastaghi, Hamid R. Dalili Oskouei
2012 arXiv   pre-print
Sattar, "An improved knapsack public key cryptography system", International Journal of Internet Technology and Secured Transactions, Vol.3 (3), pp.310-319, 2011] used Shamir's attack on the basic Merkle-Hellman  ...  This cryptosystem uses a super-increasing sequence as private key and the authors investigate a new algorithm called permutation combination algorithm to enhance density of knapsack to avoid the low-density  ...  Conclusion We considered cryptanalysis of a knapsack-type public key cryptosystem.  ... 
arXiv:1210.7417v1 fatcat:leoc6jp6xza5faz6obazian67y

An improved knapsack public key cryptography system

Sattar J. Aboud
2011 International Journal of Internet Technology and Secured Transactions  
This cryptosystem uses a super-increasing sequence as private key and the authors investigate a new algorithm called permutation combination algorithm to enhance density of knapsack to avoid the low-density  ...  s cryptosystem.  ...  Introduction The first knapsack-type public key cryptosystem (PKC) was introduced by Merkle and Hellman [12] .  ... 
doi:10.1504/ijitst.2011.041298 fatcat:afb7k7okezfbvlp6ugb3lzeblm

Quadratic compact knapsack public-key cryptosystem

Baocang Wang, Yupu Hu
2010 Computers and Mathematics with Applications  
It is shown that the proposed cryptosystem is also secure against some brute-force attacks and some known key-recovery attacks including the simultaneous Diophantine approximation attack and the orthogonal  ...  In this paper, additive knapsack-type public-key cryptography is reconsidered. We propose a knapsack-type public-key cryptosystem by introducing an easy quadratic compact knapsack problem.  ...  Introduction Knapsack cryptography [1] is an important class of public-key cryptosystems in the area of public-key cryptography.  ... 
doi:10.1016/j.camwa.2009.08.031 fatcat:bvwvdczchvftzdmjwpmuzl4bhy

PKCHD: Towards A Probabilistic Knapsack Public-Key Cryptosystem with High Density

Yuan Ping, Baocang Wang, Shengli Tian, Jingxian Zhou, Hui Ma
2019 Information  
By introducing an easy knapsack-type problem, a probabilistic knapsack-type public key cryptosystem (PKCHD) is proposed. It uses a Chinese remainder theorem to disguise the easy knapsack sequence.  ...  Meanwhile, it can use the hardest knapsack vector as the public key if its density evaluates the hardness of a knapsack instance.  ...  Acknowledgments: The authors would like to thank the anonymous reviewers for their carefulness and patience, and thank Sheng Tong for the proof of Theorem 8 and Fagen Li for paper preparation.  ... 
doi:10.3390/info10020075 fatcat:bbs55j7ovjfslph6yxojd4saju

LATTICE BASED TOOLS IN CRYPTANALYSIS FOR PUBLIC KEY CRYPTOGRAPHY

R. Santosh Kumar, C. Narasimham, S. Pallam Setty
2012 Zenodo  
Not only the existence of lattice based cryptosystems of hard in nature, but also has vulnerabilities by lattice reduction techniques.  ...  In this survey paper, we are focusing on point lattices and then describing an introduction to the theoretical and practical aspects of lattice reduction.  ...  Algorithms for finding simultaneous Diophantine approximation have been used to break some knapsack public key cryptosystems. is said to be a simultaneous Diophantine approximation of ߜ-quality to the  ... 
doi:10.5281/zenodo.3351415 fatcat:rttkitq545cs5aio7nsgg4cmfu

Diophantine Approximation Attack on a Fast Public Key Cryptosystem [chapter]

Wang Baocang, Hu Yupu
2006 Lecture Notes in Computer Science  
Yoo etc. proposed a public key cryptosystem using matrices over a ring, which was analyzed using lattice basis reduction algorithms by Youssef etc. at ACISP 2001.  ...  It is shown that the decryption of the cryptosystem can be transformed into solving the simultaneous Diophantine approximation problem, which can be approximated by lattice basis reduction algorithms.  ...  At ACISP 2000, Yoo etc. proposed a fast public key cryptosystem using matrices over a ring [9] .  ... 
doi:10.1007/11689522_3 fatcat:wfympgoumbcinhdfkelr2mop54

Cryptanalysis of a New Knapsack Type Public-Key Cryptosystem [article]

Roohallah Rastaghi
2013 arXiv   pre-print
Recently, Hwang et al. introduced a knapsack type public-key cryptosystem. They proposed a new algorithm called permutation combination algorithm.  ...  We show that this cryptosystem is not secure, as it based on basic Merkel-Hellman knapsack cryptosystem and because of the superincreasing structure, we can use shamir's attack on the basic Merkel-Hellman  ...  attack [14] and diophantine approximation attack [17] .  ... 
arXiv:1210.8375v2 fatcat:tneywny3dff43aemubglidr6sy

Performance analysis of Shamir's attack on the basic Merkle-Hellman knapsack cryptosystem [chapter]

J. C. Lagarias
1984 Lecture Notes in Computer Science  
Let R = maximum # cipher text bits # plain text bits _ ________________________ denote the rate at which a knapsack cryptosystem transmits information, and let n denote the number of items in a knapsack  ...  This paper gives a performance analysis of one variant of Shamir's attack on the basic Merkle-Hellman knapsack cryptosystem, which we call Algorithm S.  ...  Introduction In 1978 Merkle and Hellman [11] proposed public key cryptosystems based on the knapsack problem. The simplest of these cryptosystems, the basic knapsack cryptosystem, works as follows.  ... 
doi:10.1007/3-540-13345-3_28 fatcat:2jvpnnb3tnb2lhmcnvawesvnoa

Preimage Selective Trapdoor Function: How to Repair an Easy Problem

Baocang Wang
2014 Mathematical Problems in Engineering  
Public key cryptosystems are constructed by embedding a trapdoor into a one-way function. So, the one-wayness and the trapdoorness are vital to public key cryptography.  ...  The compact knapsack problem is used to construct a probabilistic public key cryptosystem, the underlying encryption function of which is proven to be preimage selective trapdoor one-way functions under  ...  See [6, 7] for more details about the relation between the simultaneous Diophantine approximation problem and knapsack public key cryptanalytics.  ... 
doi:10.1155/2014/475678 fatcat:3is4rruufrcyhfqhtc5kgiz35q

Page 4386 of Mathematical Reviews Vol. , Issue 86i [page]

1986 Mathematical Reviews  
The basic Merkle- Hellman additive trapdoor knapsack public-key cryptosystem has already been shown to be insecure by many re- searchers.  ...  The corresponding public key is A,m with A=T'T. A signature y of a message z € Z,, is any y €(Z,,)” such that y’ Ay approximates z, e.g. |z — y" Ay| < 4m?”".  ... 

A new public-key cipher system based upon the diophantine equations

C.H. Lin, C.C. Chang, R.C.T. Lee
1995 IEEE transactions on computers  
Index Terms-Public keys, private keys, cryptosystems, Diophantine equation problems, integer knapsack problems, one-way functions, trapdoor one-way functions, NP-complete.  ...  A b~t r~~t - A new public-key (two-key) cipher scheme is proposed in this paper. In our scheme, keys can be easily generated. In addition, both encryption and decryption procedures are simple.  ...  ACKNOWLEDGMENT The authors thank the referees for many helpful suggestions and comments. 1989  ... 
doi:10.1109/12.368013 fatcat:w6xo3miz2bgfzarntvkqagprya

Cryptanalysis of a quadratic knapsack cryptosystem

Amr M. Youssef
2011 Computers and Mathematics with Applications  
Wang and Hu [B. Wang and Y. Hu, Quadratic compact knapsack public-key cryptosystem, Comput. Math.  ...  Appl. 59 (1) (2010) 194-206] proposed a knapsack-type public-key cryptosystem by introducing an easy quadratic compact knapsack problem and then using the Chinese remainder theorem to disguise the easy  ...  Introduction Knapsack-based cryptosystems [1, 2] were among the first public-key systems to be invented.  ... 
doi:10.1016/j.camwa.2011.01.006 fatcat:d76l4d4n6fcofhe2gr3u73fpsq

Page 3308 of Mathematical Reviews Vol. , Issue 86g [page]

1986 Mathematical Reviews  
Theory 30 (1984), no. 5, 699-704], who showed how to break the basic Merkle-Hellman knapsack public key cryptosystem [R. C. Merkle and M. E. Hellman, ibid. 24 (1978), no. 5, 525-530].  ...  He then goes on to discuss Shamir’s attack on the basic Merkle- Hellman cryptosystem which derives a set of Diophantine approximation inequalities and solves them using the author’s integer programming  ... 
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