## 1 Introduction  
Wireless sensor networks (WSNs) are wireless networks consisting of a large number of resource-constrained sensor nodes, where each node is equipped with a sensor to monitor physical phenomena, such as temperature, light, and pressure. The main features of WSNs are resource constraints, such as storage, computing power, and sensing distance. Recently, the energy consumption of data centers has attracted attention because of the fast growth of data throughput. WSNs can provide a solution for data collection and data processing in various applications including data center monitoring. That is, WSNs can be utilized for data center monitoring to improve the efficiency of energy consumption. Several solutions were proposed to solve this problem [1; 2]. Since sensor nodes are usually deployed in remote areas and left unattended, they can be led to network security issues, such as node capture, eavesdropping, and message tampering during data communication. Additionally, many application areas of WSNs require data confidentiality, integrity, authentication, and non-repudiation, meaning there is a need for an efficient cryptographic mechanism to satisfy current security requirements. However, due to the constraint of WSNs, it is difficult to utilize the conventional cryptographic algorithms. Therefore, efficient cryptographic algorithms considering code size, computation time, and power consumption are required for the security of WSNs. In 1985, elliptic curve cryptography (ECC) was proposed independently of public key cryptography (PKC) by [PERSON] and [PERSON] [3; 4]. ECC is mainly used for digital signature and key exchange based on the elliptic curve discrete logarithm problem (ECDLP), which is defined by elliptic curve point operations in a finite field. ECC provides the same security level with a smaller key size compared to existing PKC algorithms such as Rivest-Shamir-Adleman (RSA) cryptosystem [5]. For example, ECC over \(\mathbb{F}_{p}\) with a 256-bit prime \(p\) provides an equivalent security level as RSA using 3072-bit key. Because RSA uses a small integer as the public key, RSA public key operations can be efficiently computed. However, RSA private key operations are extremely slower than ECC, therefore they have limited use in the applications of WSNs. Therefore, ECC can be efficiently utilized than RSA for resource-constrained WSNs devices, such as smart cards and sensor nodes. However, recently proposed manipulation and backdoors have raised the suspicion of weakness in previous ECC standards. In particular, the National Institute of Standards and Technology (NIST) P-224 curve is not secure against twist attacks, which are the combined attacks that use the small-subgroup attacks and the invalid-curve attacks using the twist of curve [6]. The dual elliptic curve deterministic random bit generator (Dual_EC_DRBG) is a pseudo-random number generator (PRNG) standardized in NIST SP 800-90A. However, the revised version of NIST SP 800-90A standard removes Dual_EC_DRBG because this algorithm contains a backdoor for the national security agency (NSA) [7]. Therefore, the demand for next generation elliptic curves has increased. Specific examples of such curves are Curve25519, Ed448-Goldilocks, and twisted Edwards curves [8; 9; 10]. The main features of these curves are the selection of efficient parameters. The Curve25519 utilizes a prime of the form \(p=2^{255}-19\) and a fast Montgomery elliptic curve. The Ed448-Goldilocks curve utilizes a Solinas trinomial prime of the form \(p=2^{448}-2^{224}-1\), which provides fast field arithmetic on both 32-bit and 64-bit machines because \(224=28\times 8=32\times 7=56\times 4\). These parameters can accelerate the performance of ECC-based protocols. The details of the twisted Edwards curves can be found in Section 2.3. Scalar multiplicationor point multiplication computes an operation \(kP\) using an elliptic curve point \(P\) and a scalar \(k\). This operation determines the performance of ECC.