Locally repairable codes are widely applicable in contemporary large-scale distributed cloud storage systems and various other areas. By making use of some algebraic structures of elliptic curves, Li et al. developed a series ofq-ary optimal locally repairable codes with lengths that can extend toq+2 √q. In this paper, we generalize their methods to hyperelliptic curves of genus 2, resulting in the construction of several new families ofq-ary optimal or almost optimal locally repairable codes. Our codes feature lengths that can approachq+ 4 √q, and the locality can reach up to 239. Furthermore, we prove that our locally repairable codes are optimal when the localityrand the minimum distancedsatisfyr+ 1 =d∈ {3, 4} orr=d= 4.

In isogeny‐based cryptography, bilinear pairings are regarded as a powerful tool in various applications, including key compression, public key validation, and torsion basis generation. However, in most isogeny‐based protocols, the performance of pairing computations is unsatisfactory due to the high computational cost of the Miller function. Reducing the computational expense of the Miller function is crucial for enhancing the overall performance of pairing computations in isogeny‐based cryptography. This paper addresses this efficiency bottleneck. To achieve this, we propose several techniques for a better implementation of pairings in isogeny‐based cryptosystems. We use (modified) Jacobian coordinates and present new algorithms for Miller function computations to compute pairings of order 2 ∙ and 3 ∙ . For pairings of arbitrary order, which are crucial for key compression in some SIDH‐based schemes (such as M‐SIDH and binSIDH), we combine Miller doublings with Miller additions/subtractions, leading to a considerable speedup. Moreover, the optimizations for pairing applications in CSIDH‐based protocols are also considered in this paper. In particular, our approach for supersingularity verification in CSIDH is 15.3% faster than Doliskani’s test, which is the state‐of‐the‐art.

Isogeny-based cryptography is famous for its short key size. As one of the most compact digital signatures, SQIsign (Short Quaternion and Isogeny Signature) is attractive among post-quantum cryptography, but it is inefficient compared to other post-quantum competitors because of complicated procedures in the ideal-to-isogeny translation, which is the efficiency bottleneck of the signing phase. In this paper, we recall the current implementation of SQIsign and mainly focus on how to improve the execution of the ideal-to-isogeny translation in SQIsign. Specifically, we demonstrate how to utilize the reduced Tate pairing to save one of the two elliptic curve discrete logarithms. In addition, the efficient implementation of the remainder discrete logarithm computation is explored. We speed up other procedures in the ideal-to-isogeny translation with various techniques as well. It should be noted that our improvements also benefit the performance of key generation and verification in SQIsign. In the instantiation with$ {p_{1973}}$, the improvements lead to a speedup of 5.47%, 8.80% and 25.34% for key generation, signature and verification, respectively.

In recent years, the isogeny-based protocol, namely supersingular isogeny Diffie-Hellman (SIDH) has become highly attractive for its small public key size. In addition, one can utilize several techniques to further compress the public key. However, compared to other post-quantum protocols, the computational cost of SIDH is relatively high, and so is that of its public-key compression. On the other hand, the storage for pairing computation and discrete logarithms to speed up the current implementation of the key compression is somewhat large. In this paper, we mainly improve the performance of public-key compression of SIDH, especially the efficiency and the storage of pairing computation involved. Our experimental results show that the memory requirement for pairing computation is reduced by a factor of about 1.5. Meanwhile, the instantiation of public-key compression of SIDH is$6.95\%--10.44\%$faster than the current state-of-the-art. Although SIKE is broken now, the techniques in this paper may benefit other isogeny-based cryptosystems which are still secure.

We first introduce a family of binary pq2-periodic sequences based on the Euler quotients modulo pq, where p and q are two distinct odd primes and p divides q - 1. The minimal polynomials and linear complexities are determined for the proposed sequences provided that 2q-1≠ 1 mod q2. The results show that the proposed sequences have high linear complexities.

Here, the authors provide a derivation of an improvement to Kanayama's elliptic curve scalar multiplication algorithm using division polynomials. In addition, they also provide experimental results and show that the improvement is useful when the elliptic curve under consideration is defined over large prime fields.

The binary threshold sequence with period 2p2is defined from the Euler quotient modulo 2p. The trace presentation of its subsequence is devised and then the linear complexity of the original sequence can be determined efficiently.

Scalar multiplication is the most important and expensive operation in elliptic curve cryptosystems. In this study, the authors improve the efficiency of the elliptic net algorithm to compute scalar multiplication by using the equivalence of elliptic nets. The proposed method saves four multiplications by a constant in each iteration loop. Experimental results also indicate that the proposed algorithm will be more efficient than the previously known results on this line while it is still slower than the state‐of‐the‐art algorithm to compute scalar multiplication.

In this paper we propose a modified Elliptic Net algorithm to compute pairings. By reducing the number of the intermediate variables which should be updated in the iteration loop of the Elliptic Net algorithm, we speed up the computation of pairings. Experimental results show that the proposed method is more efficient than the original Elliptic Net algorithm on Barreto-Naehrig (BN) curves which are the best choice for implementing pairings at 128-bit security level.

We investigate multi-point algebraic geometric codes defined from curves related to the generalized Hermitian curve introduced by Bassa et al. Our main result is to find a basis of the Riemann-Roch space of a series of divisors, which can be used to construct multi-point codes explicitly. These codes turn out to have nice properties similar to those of Hermitian codes, for example, they are easy to describe, to encode and decode. It is shown that the duals are also such codes and an explicit formula is given. In particular, this formula enables one to calculate the parameters of these codes. Finally, we apply our results to obtain linear codes attaining new records on the parameters. A new record-giving [234, 141,≥ 59]-code over F27is presented as one of the examples.

Self-pairings have found interesting applications in cryptographic schemes. In this paper, we present a novel method for constructing a self-pairing on supersingular elliptic curves with even embedding degrees, which we call the Ateil pairing. This new pairing improves the efficiency of the self-pairing computation on supersingular curves over finite fields with large characteristic. Based on the ηTpairing, we propose a generalization of the Ateil pairing, which we call the Ateilipairing. The optimal Ateilipairing which has the shortest Miller loop is faster than previously known self-pairings on supersingular elliptic curves over finite fields with small characteristic. We also present a new self-pairing based on the Weil pairing which is faster than the self-pairing based on the Tate pairing on ordinary elliptic curves with embedding degreeone.