Cryptanalysis of Merkle-Hellman Cipher Using Parallel Genetic Algorithm

  • Nedjmeddine KantourEmail author
  • Sadek Bouroubi


In 1976, Whitfield Diffie and Martin Hellman introduced the public key cryptography or asymmetric cryptography standards. Two years later, an asymmetric cryptosystem was published by Ralph Merkle and Martin Hellman called \( \mathcal {M} \mathcal {H} \), based on a variant of knapsack problem known as the subset-sum problem which is proven to be NP -hard. Furthermore, over the last four decades, Metaheuristics have achieved remarkable progress in solving NP-hard optimization problems. However, the conception of these methods raises several challenges, mainly the adaptation and the parameters setting. In this paper, we propose a Parallel Genetic Algorithm (PGA) adapted to explore effectively the search space of considerable size in order to break the \( \mathcal {M} \mathcal {H} \) cipher. Experimental study is included, showing the performance of the proposed attacking scheme, and finally a concluding comparison with lattice reduction attacks.


Cryptanalysis Merkle-Hellman Cryptosystem Knapsack problem Genetic algorithm Lattice reduction algorithms 



The authors thank the anonymous referees for their valuable time to review our manuscript. This work was supported by L’IFORCE Laboratory.


  1. 1.
    Hellman M, Merkle R (1978) Hiding information and signatures in trapdoor knapsacks. IEEE Trans Inf Theory 24(5):525–530CrossRefGoogle Scholar
  2. 2.
    Karp R (1972) Reducibility among combinatorial problems. In: Miller RE, Thatcher JW (eds) Complexity of computer computations. Plenum Press, New York, pp 85–103Google Scholar
  3. 3.
    Reeves CR, Rowe JE (2003) Genetic algorithms - principles and perspectives: a guide to GA theory. Kluwer Academic Publishers, DordrechzbMATHGoogle Scholar
  4. 4.
    Shamir A (1984) A polynomial time algorithm for breaking the basic Merkle Hellman cryptosystem. IEEE Trans Inf Theory IT-30(5):699–704MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Spillman R (1993) Cryptanalysis of knapsack ciphers using genetic algorithms. Cryptologia 17(4):367–377CrossRefzbMATHGoogle Scholar
  6. 6.
    Lenstra AK, Lenstra HW Jr~, Lovász L (1982) Factoring polynomials with rational coefficients. Math Ann, pp 515–534Google Scholar
  7. 7.
    Sinha S, Palit S, Molla M, Khanra A, Kule M (2011) A cryptanalytic attack on knapsack cipher using differential evolution algorithm, recent advances in intelligent computational systems (RAICS), IEEEGoogle Scholar
  8. 8.
    Goldberg DE (1989) Genetic algorithms in search, optimization and machine learning. Addison-Wesley, ReadingzbMATHGoogle Scholar
  9. 9.
    Melanie M (1996) An introduction to genetic algorithms. MIT Press, CambridgezbMATHGoogle Scholar
  10. 10.
    Holland J (1975) Adaption in natural and artificial systems, ann arbor MI. The University of Michigan Press, MichiganGoogle Scholar
  11. 11.
    Sivanandam SN, Deepa SN (2008) Introduction to genetic algorithms. Springer, BerlinzbMATHGoogle Scholar
  12. 12.
    Stamp M (2005) Information security: principles and practice. Wiley-InterscienceGoogle Scholar
  13. 13.
    McAndrew A (2011) Introduction to cryptography with open-source software. CRC Press, Boca Raton, Florida, USAzbMATHGoogle Scholar
  14. 14.
    Kreher DL, Stinson DR (1999) Combinatorial Algorithms: generation, enumeration and search. CRC Press, Boca Raton, Florida, USAzbMATHGoogle Scholar
  15. 15.
    Lagrias JC, Odlyzko AM (1985) Solving low-density subset problems. J ACM (JACM) 32(1):229–246MathSciNetCrossRefGoogle Scholar
  16. 16.
    Coster MJ, Joux A, LaMacchia BA, Odlyzko AM, Schnorr C, Stern J (1992) An improved low-density subset sum algorithm. Computational Complexity, 2Google Scholar
  17. 17.
    Adleman LM (1983) On beaking generalized knapsack public key cryptosystems. In: ACM Proceedings of 15th STOCGoogle Scholar
  18. 18.
    Stein W et al (2009) Sage mathematics software (Version 4.2.1). The sage development team 14th.
  19. 19.
    Palit S, Sinha S, Molla M, Khanra A, Kule M (2011) A cryptanalytic attack on the knapsack cryptosystem using binary firefly algorithm. In: 2nd international conference on computer and communication technology (ICCCT). IEEE, pp 428–432Google Scholar
  20. 20.
    Mandal T, Kule M (2016) An improved cryptanalysis technique based on Tabu search for Knapsack cryptosystem. Int J Control Theory Appl 16(9):8295–8302Google Scholar
  21. 21.
    Garg P, Shastri A, Agarwal DC (2007) An enhanced cryptanalytic attack on Knapsack Cipher using genetic algorithm. World academy of science, engineering and technology, international science index 12. Int J Comput Electrical Automation Control Inf Eng 1(12):4071–4074Google Scholar
  22. 22.
    Jain A, Chaudhari NS (2014). In: de la Puerta J et al (eds) International joint conference SOCO14-CISIS14-ICEUTE14 advances in intelligent systems and computing, vol 299. Springer, ChamGoogle Scholar
  23. 23.
    Abdel-Basset M, El-Shahat D, El-henawy I, Sangaiah AK, Ahmed SH (2018) A novel whale optimization algorithm forcryptanalysis in Merkle-Hellman cryptosystem. Mobile Netw Appl 23(4): 1–11CrossRefGoogle Scholar
  24. 24.
    Schnorr CP, Shevchenko T (2012) Solving subset sum problems of density close to 1 by randomized BKZ-reduction. IACR Cryptology ePrint Archive 2012:620Google Scholar
  25. 25.
    Howgrave-Graham N, Joux A (2010) New generic algorithms for hard knapsacks. In: Gilbert H (ed) Advances in cryptology – EUROCRYPT 2010. EUROCRYPT 2010. Lecture notes in computer science, vol 6110. Springer, BerlinGoogle Scholar
  26. 26.
    Koiliaris K, Xu C (2017) A faster pseudopolynomial time algorithm for subset sum. In: SODA17Google Scholar
  27. 27.
    Bringmann K (2017) A near-linear pseudopolynomial time algorithm for subset sum. In: Proceedings of the 28th annual ACM-SIAM symposium on discrete algorithms, pp 1073–1084Google Scholar
  28. 28.
    Jen SM, Lai TL, Lu CY, Yang JF (2012) Knapsack cryptosystems and unreliable reliance on density. In: 2012 IEEE 26th international conference on advanced information networking and applications (AINA). IEEE, pp 748–754Google Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.L’IFORCE Laboratory, Faculty of MathematicsUniversity of Sciences and Technology Houari Boumediene (USTHB)AlgiersAlgeria

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