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New designing of cryptosystems based on quadratic fields

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Abstract

This paper proposes a method to construct new kind of non-maximal imaginary quadratic order (NIQO*) by combining the technique of Diophantine equation and the characters of non-maximal imaginary quadratic order. It is proved that in the class group of this new kind of NIQO*, it is very easy to design provable secure cryptosystems based on quadratic field (QF). With the purpose to prove that this new kind of QF-based cryptosystems are easy to implement, two concrete schemes are presented, i.e., a Schnorr-like signature and an ElGamel-like encryption, by using the proposed NIQO*. In the random oracle model, it is proved that: (1) under the assumption that the discrete logarithm problem over class groups (CL-DLP) of this new kind of NIQO* is intractable, the proposed signature scheme is secure against adaptive chosen-message attacks, i.e., achieving UF-CMA security; (2) under the assumption that the decisional Diffie-Hellman problem over class groups (CL-DDH) of this new kind of NIQO* is intractable, the enhanced encryption in this paper is secure against adaptive chosen-ciphertext attacks, i.e., reaching IND-CCA2 security.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, China

    XiaoLei Dong, ZhenFu Cao & LiCheng Wang

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  1. XiaoLei Dong
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  2. ZhenFu Cao
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  3. LiCheng Wang
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Corresponding author

Correspondence to ZhenFu Cao.

Additional information

Supported by the National Natural Science Foundation of China (Grant Nos. 60673079, 60773086 and 60572155)

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Dong, X., Cao, Z. & Wang, L. New designing of cryptosystems based on quadratic fields. Sci. China Ser. F-Inf. Sci. 51, 1106–1116 (2008). https://doi.org/10.1007/s11432-008-0091-4

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