Hardness Of Discrete Logarithm Problem Research Articles

Constructing CM Fields for NFS to Accelerate DL Computation in Non-Prime Finite Fields

The hardness of discrete logarithm problem (DLP) over finite fields is the security foundation of many cryptographic protocols. When the characteristic is not small, the state-of-the-art algorithms for solving DLP are the number field sieve (NFS) and its variants. In the relation collection step, to translate the relations between prime ideals to those of elements one needs to use the Schirokauer map. Besides, if the number field has non-trivial automorphisms, one can use them to accelerate the factor-base logarithms computation. However, the Schirokauer map is not compatible with automorphisms. To exploit automorphism efficiently, we focus on the method to construct fields in NFS such that the fields on both sides have non-trivial automorphisms with the logarithms of units being zero. 1. We construct two families of CM polynomials of arbitrary even degree with small coefficients, corresponding to the automorphisms being <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ↦ – <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> or <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> ↦ 1/ <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</i> . 2. We show how to combine these polynomials with the JLSV <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> and Conjugation polynomial selection methods on both sides. 3. We also generalize our method to the multiple number field sieve and the extended tower number field sieve.

An Efficient Certificate-Based Aggregate Signature Scheme With Provable Security for Industrial Internet of Things

In Industrial Internet of Things (IIoT), the certificate-based aggregate signature (CBAS) scheme was used to ensure privacy and security of IIoT, thus, the concrete construction of CBAS scheme without bilinear pairing has proposed, which was employed as an underlying building block to create an efficient data aggregation protocol. In this article, we first point out that the previous CBAS scheme cannot keep its claimed security. Particularly, a malicious KGC can compute a valid forgery signature, because it has the knowledge of the master secret key. After that, we discussed how specific forgery attacks could be made to against the existing CBAS schemes, which demonstrates that the previous related constructions cannot achieve its claimed security. In order to further provide a secure CBAS scheme for IIoT, we suggest a method to improve the CBAS scheme and propose a concrete construction of it. In the random oracle model, based on the hardness of discrete logarithm problem, the security of our new proposal can be proved. Compared with the previous CBAS schemes, our construction has better security, which is efficient with respect to computation and communication.

Refined analysis to the extended tower number field sieve

The hardness of discrete logarithm problem over finite fields is the security foundation of many cryptographic protocols. When the characteristic of the finite field is medium or large, the state-of-art algorithms for solving the corresponding problem are the number field sieve and its variants. In 2016, Kim and Barbulescu presented the extended tower number field sieve, which achieves a new complexity in the medium prime case and imposes a new estimation of the security of concrete parameters in certain cryptosystems such as pairing-based cryptosystems. In this paper, a refined analysis to this algorithm is given as follows.–Firstly, a uniform formula is given for the total complexity of the extended tower number field sieve. For a given polynomial selection method, this formula can directly give the complexity in this case.–Then, a method is proposed to improve the computation in the smoothing phase by exploring subfield structures when the extension degree is composite.–At last, the complexity of the descent phase is analyzed when sieving over degree-one polynomials and high-degree polynomials respectively and it is shown still negligible compared to the improved smoothing phase.

An Efficient Threshold Verifiable Multi-Secret Sharing Scheme Using Generalized Jacobian of Elliptic Curves

‎In a (t,n)-threshold secret sharing scheme‎, ‎a secret s is distributed among n participants such that any group of t or more participants can reconstruct the secret together‎, ‎but no group of fewer than t participants can do‎. In this paper, we propose a verifiable (t,n)-threshold multi-secret sharing scheme based on Shao and Cao‎, ‎and the intractability of the elliptic curve discrete logarithm problem (ECDLP) by using generalized Jacobian of elliptic curves‎. ‎The proposed scheme has all the benefits of Shao and Cao‎, ‎however‎, ‎our scheme no need to a secure channel‎. ‎Furthermore‎, ‎we exploit the techniques via elliptic curves to perform the scheme‎. ‎This can be very important‎ ‎because the hardness of discrete logarithm problem on elliptic curves increases security of the proposed scheme‎.

Structure and Analysis of Certificateless Proxy Blind Signature Scheme without Bilinear Pairing

Combined with certificateless public key cryptography and proxy blind signature, an efficient certificateless proxy blind signature scheme is proposed. Its security is based on the discrete logarithm problem. Compared with the existed certificateless proxy blind signature scheme, because without bilinear pairing, it have higher efficiency. According to the different attacker and all kinds of attacks, the scheme is proved to be correct and security under the hardness of discrete logarithm problem in the finite field.

Certificateless Signcryption Scheme Without Bilinear Pairing

Only six certificateless signcryption schemes have been proposed in recent years.Most of them cannot provide confidentiality and authentication.Even if some of them are secure,all of them need pairing operations.In order to solve the above-mentioned problems,a pairing-free certificateless signcryption scheme is proposed,and its security has proven to be in the random oracle model(ROM) under the computational Diffie-Hellman(CDH) assumption and the hardness of discrete logarithm problem(DLP).This scheme eliminates pairing operations and is the most efficient certificateless signcryption scheme.