Abstract
In 2018, the longest vector problem and closest vector problem in local fields were introduced, as the \(p\)-adic analogues of the shortest vector problem and closest vector problem in lattices of Euclidean spaces. They are considered to be hard and useful in constructing cryptographic primitives, but no applications in cryptography were given. In this paper, we construct the first signature scheme and public-key encryption cryptosystem based on \(p\)-adic lattice by proposing a trapdoor function with the norm-orthogonal basis of \(p\)-adic lattice. These cryptographic schemes have reasonable key size and the signature scheme is efficient, while the encryption scheme works only for short messages, which shows that \(p\)-adic lattice can be a new alternative to construct cryptographic primitives and well worth studying.
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References
J. W. S. Cassels, Local Fields (Cambridge University Press, Cambridge, 1986).
Y. Deng, L. Luo and G. Xiao, ‘On some computational problems in local fields,” Cryptology ePrint Archive, Report 2018/1229, http: //eprint.iacr.org/2018/1229 (2018).
Y. Deng, L. Luo, Y. Pan and G. Xiao, “On some computational problems in local fields,” J. Syst. Sci. Comp. 35, 1191–1200 (2022).
W. Diffie and M. E. Hellman, “New directions in cryptography,” IEEE Trans. Inf. Theo. 22, 644–654 (1976).
L. Ducas and P. Nguyen, “Learning a zonotope and more: cryptanalysis of NTRUSign countermeasures,” In: X. Wang and K. Sako (Eds.), ASIACRYPT 2012, LNCS 7658, 433–450 (2012).
K. Eisenträger, S. Hallgren, K. Lauter, T. Morrison and C. Petit, “Supersingular isogeny graphs and endomorphism rings: reductions and solutions,” In: J.B. Nielsen and V. Rijmen (Eds.), EUROCRYPT 2018, LNCS 10822, 329–368 (2018).
T. ElGamal, “A public key cryptosystem and a signature scheme based on discrete logarithms,” IEEE Trans. Inf. Theo. 31, 469–472 (1985).
O. Goldreich, S. Goldwasser and S. Halevi, “Public-key cryptosystems from lattice reduction problems,” In: B. S. Kaliski Jr. (Ed.), Adv. Crypt.- CRYPTO’97, LNCS 1294, 112–131 (1997).
N. Koblitz, \(p\)-Adic Numbers, \(p\)-Adic Analysis, and Zeta-Functions, Second edition (Springer, New York, 1984).
N. Koblitz, “Elliptic curve cryptosystems,” Math. Comput. 48, 203–209 (1987).
N. Koblitz, “Hyperelliptic cryptosystems,” J. Cryptology 1, 139–150 (1989).
N. Koblitz, Algebraic Aspects of Cryptography (Springer, Berlin, 1998).
D. Kohel, Endomorphism Rings of Elliptic Curves over Finite Fields, Ph.D. thesis (University of California, Berkeley, 1996).
T. Matsumoto and H. Imai, “Public quadratic polynomial-tuples for efficient signature-verification and message-encryption,” In: C. G. Guenther (Ed.), Adv. Crypt. - EUROCRYPT’88, LNCS 330, 419–453 (1988).
R. J. McEliece, “A public-key cryptosystem based on algebraic coding theory,” DSN Prog. Rep. 42-44, 114–116 (Jet Propulsion Laboratory, 1978).
D. Micciancio and S. Goldwasser, Complexity of Lattice Problems, A Cryptographic Perspective (Kluwer, Boston, 2002).
V. S. Miller, “Use of elliptic curves in cryptography,” In: H.C. Williams (Ed.), Adv. Crypt. - CRYPTO’85, LNCS 218, 417–426 (1986).
W. Narkiewicz, Elementary and Analytic Theory of Algebraic Numbers, Third edition (Springer, New York, 2004).
P. Nguyen, “Cryptanalysis of the Goldreich-Goldwasser-Halevi cryptosystem from Crypto’97,” In: M. Wiener (Ed.), Adv. Crypt. - CRYPTO’99, LNCS 1666, 288–304 (1999).
P. Nguyen and O. Regev, “Learning a parallelepiped: cryptanalysis of GGH and NTRU signatures,” In: S. Vaudenay (Ed.), EUROCRYPT 2006, LNCS 4004, 271–288 (2006).
R. L. Rivest, A. Shamir and L. Adleman, “A method for obtaining digital signatures and public key cryptosystems,” Comm. ACM, 21, 120–126 (1978).
J.-P. Serre, Local Fields (Springer, New York, 1979).
P. W. Shor, “Algorithms for quantum computation: discrete logarithms and factoring,” in Proc. 35th Annual Symposium on Foundations of Computer Science, IEEE Comp. Soc. Press, pp. 124–134 (Los Alamitos, CA, 1994).
C. L. Siegel, Lectures on the Geometry of Numbers (Springer, Berlin, 1989).
A. Weil, Basic Number Theory, Third edition (Springer, New York, 1974).
https://csrc.nist.gov/Projects/Post-Quantum-Cryptography.
Funding
This work was supported by National Natural Science Foundation of China (No. 12271517) and National Key Research and Development Project of China (No. 2018YFA0704705).
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Deng, Y., Luo, L., Pan, Y. et al. Public-Key Cryptosystems and Signature Schemes from \(p\)-Adic Lattices. P-Adic Num Ultrametr Anal Appl 16, 23–42 (2024). https://doi.org/10.1134/S2070046624010035
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DOI: https://doi.org/10.1134/S2070046624010035