# A Crypto Accelerator of Binary Edward Curves for Securing Low-Resource Embedded Devices
## 1 Introduction  
With the widespread use of the internet as a communication medium, there is a growing need for effective solutions to protect information across various devices [1]. In this context, cryptography is a technique that can safeguard sensitive data on an insecure medium [2]. Therefore, two fundamental approaches emerge, known as symmetric and asymmetric encryption. The former relies on a shared key for both encryption and decryption. However, the management of keys within large organizations poses a formidable challenge [3]. The latter, also referred to as public key cryptography, is based on two keys: The first key, known as the public key, is used for encryption. The second key, known as the private key, is used for decryption [4].  
Among the prominent pioneers in the field of asymmetric cryptography are RSA ([PERSON], [PERSON], and [PERSON]) [5] and ECC (elliptic curve cryptography) [6]. The security prowess of RSA stems from the formidable difficulty of factoring in large numbers. On the other hand, ECC harnesses the fascinating mathematics of elliptic curves. Notably, ECC keys offer a significant advantage over RSA keys in terms of length. Despite their shorter size, a 256-bit ECC key can provide equivalent security to a substantial 3072-bit RSA key [7]. This fact positions ECC as an enticing alternative, particularly for resource-constrained applications, like IoT (internet of things) [8, 9]. From the implementation point of view, ECC involves foundational arithmetic operations. These arithmetic operations support essential group operations. As a result, cryptographic computations are performed, facilitating protocol functions like key exchange, digital signatures, and encryption.  
A noteworthy consideration is the vulnerability of ECC to simple power analysis (SPA) attacks [10]. An SPA attack belongs to the category of a side-channel attack. It capitalizeson the information leaked by the physical implementation of an encryption algorithm [11]. In ECC, the computationally intensive operation of point multiplication is particularly susceptible to SPA attacks. This vulnerability underscores the importance of implementing countermeasures. To address this issue, the fixed window algorithm has been specifically designed. In addition to the fixed window algorithm, the unified mathematical formulas for ECC also help to reduce the risk of SPA attacks.  
Some examples of unified mathematical formulas for ECC include binary Edwards curves (BECs) [12], binary Huff curves (BHCs) [13], and Hessian curves (HCs) [14]. The choice of particular mathematical formulas depends on the specific requirements of the application and the trade-off between performance and security. For example, BHC and BEC are defined over binary fields and are considered more appropriate for area-constrained applications [15; 16]. HC, on the other hand, is defined over higher-dimensional fields and offers a higher resistance to side-channel attacks. However, the increased resistance comes at the cost of increased complexity and reduced efficiency. In general, the BEC is considered to be a good choice for resource-constrained devices, as it offers a good balance of security, efficiency, and simplicity [16]. Some of the applications requiring high throughput with a smaller area include financial transactions [17], cloud computing [18], blockchain [19; 20], smart cards [21] and wireless communication [22]. By optimizing hardware for BEC, the operations are performed securely at a higher speed using minimal hardware cost.