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Abstract
Lattice-based cryptography is encryption that uses lattice issues as the basis of a conjecture of the complexity of the worst-case problem. This complexity is considered hard in the world of classical and quantum providing for a primitives design of lattice-based cryptography algorithms. The NIST held a contest to find the best algorithm to implement. This paper compares three finalists of that contest by analyzing the theoretical basis of the lattice problems they hold, with the security of the worst case of the attack through some algorithms, lattices with underlying problems, construction of cryptographic algorithms, and comparative analysis of three finalists of the lattice schemes of the NIST competition. Comparison includes the safety and robustness of the lattice problem used the cryptographic algorithm that is built and finally there will be an analysis in a computational time context to discuss the efficiency algorithm created. Two advantages and disadvantages of each the algorithm finalist are implemented and also described.
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Copyright (c) 2025 Mohammed Hasan Al-Dulaimi, Suha Kamal Al-Dulaimi (Author)

This work is licensed under a Creative Commons Attribution 4.0 International License.
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References
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- R. J. McEliece, “A public-key cryptosystem based on algebraic coding theory,” DSN Progress Report 44, Jet Propulsion Laboratory (JPL), California Institute of Technology, Pasadena, CA, USA, pp. 114–116, 1978.
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References
D. J. Bernstein, J. Buchmann, and E. Dahmen, Post-Quantum Cryptography. Berlin, Germany: Springer, 2009.
L. Chen et al., Report on Post-Quantum Cryptography, NISTIR 8105. Gaithersburg, MD, USA: National Institute of Standards and Technology, 2016.
G. Alagic et al., Status Report on the Second Round of the NIST Post-Quantum Cryptography Standardization Process, NISTIR 8309. Gaithersburg, MD, USA: National Institute of Standards and Technology, 2020.
G. Alagic et al., Status Report on the Third Round of the NIST Post-Quantum Cryptography Standardization Process, NISTIR 8413. Gaithersburg, MD, USA: National Institute of Standards and Technology, 2022.
J. Hoffstein, J. Pipher, and J. H. Silverman, An Introduction to Mathematical Cryptography, 2nd ed. New York, NY, USA: Springer, 2014.
L. Ducas and D. Micciancio, “Faster Gaussian sampling for trapdoor lattices with arbitrary modulus,” in Advances in Cryptology – CRYPTO 2015, Proc. of the 35th Annual International Cryptology Conference, Santa Barbara, CA, USA, 2015, pp. 703–728, doi: https://doi.org/10.1007/978-3-662-48000-7_34.
C. Peikert, “A decade of lattice cryptography,” Foundations and Trends in Theoretical Computer Science, vol. 10, no. 4, pp. 283–424, 2016, doi: https://doi.org/10.1561/0400000074.
N. Bindel, J. Buchmann, and J. Krämer, “Lattice-based cryptography: The role of security and performance trade-offs,” Journal of Cryptographic Engineering, vol. 9, no. 1, pp. 9–32, 2019, doi: https://doi.org/10.1007/s13389-018-0193-6.
J. W. Bos et al., “CRYSTALS–Kyber: A CCA-secure module-lattice-based KEM,” in Proc. IEEE Symposium on Security and Privacy (SP 2018), San Francisco, CA, USA, 2018, pp. 353–367, doi: https://doi.org/10.1109/SP.2018.00040.
W. Castryck et al., “CSIDH: An efficient post-quantum commutative group action,” in Advances in Cryptology – ASIACRYPT 2018, Proc. of the 24th International Conference on the Theory and Application of Cryptology and Information Security, Brisbane, Australia, 2018, pp. 395–427, doi: https://doi.org/10.1007/978-3-030-03329-3_14.
A. Hülsing et al., XMSS: Extended Merkle Signature Scheme, RFC 8391. Internet Engineering Task Force (IETF), 2018.
D. Aggarwal et al., “Quantum attacks on Bitcoin, and how to protect against them,” Ledger, vol. 2, pp. 42–61, 2017, doi: https://doi.org/10.5195/ledger.2017.46.
L. K. Grover, “A fast quantum mechanical algorithm for database search,” in Proc. 28th Annual ACM Symposium on Theory of Computing (STOC 1996), Philadelphia, PA, USA, 1996, pp. 212–219, doi: https://doi.org/10.1145/237814.237866.
P. W. Shor, “Algorithms for quantum computation: Discrete logarithms and factoring,” in Proc. 35th Annual Symposium on Foundations of Computer Science (FOCS 1994), Santa Fe, NM, USA, 1994, pp. 124–134.
J. Ding et al., “Practical transformations from ideal lattice-based cryptosystems to the ring-LWE setting,” ACM Transactions on Privacy and Security, vol. 21, no. 2, pp. 1–37, 2018, doi: https://doi.org/10.1145/3196491.
M. R. Albrecht, S. Bai, and L. Ducas, “A subfield lattice attack on overstretched NTRU assumptions,” in Advances in Cryptology – CRYPTO 2016, Proc. of the 36th Annual International Cryptology Conference, Santa Barbara, CA, USA, 2016, pp. 153–178, doi: https://doi.org/10.1007/978-3-662-53018-4_6.
R. J. McEliece, “A public-key cryptosystem based on algebraic coding theory,” DSN Progress Report 44, Jet Propulsion Laboratory (JPL), California Institute of Technology, Pasadena, CA, USA, pp. 114–116, 1978.
V. Lyubashevsky, C. Peikert, and O. Regev, “On ideal lattices and learning with errors over rings,” Journal of the ACM, vol. 60, no. 6, pp. 1–35, 2013, doi: https://doi.org/10.1145/2535925.
O. Regev, “On lattices, learning with errors, random linear codes, and cryptography,” Journal of the ACM, vol. 56, no. 6, pp. 1–40, 2009, doi: https://doi.org/10.1145/1568318.1568324.
National Institute of Standards and Technology, Post-Quantum Cryptography Standardization: Third Round Candidates. Gaithersburg, MD, USA, 2020.
