# Large Field-Size Elliptic Curve Processor for Area-Constrained Applications
## 1 Introduction  
The public-key cryptographic standards (ECC [1] and RSA [2]) are vulnerable to quantum computers when running a [PERSON]'s algorithm [3, 4]. In 2022, the National Institute of Standards and Technology (NIST) has standardized new quantum-secure key encapsulation and digital signature algorithms [5]. However, the NIST-recommended quantum algorithms are not reasonable for securing area constrained cryptographic applications such as wireless sensor nodes (WSNs) [6, 7], radio-frequency-identification-networks (RFID) [8, 9], automotive vehicles [10], secure robotics communication [11], etc. The reason is the huge hardware resource requirements due to several hash computations, binomial and gaussian samplers for introducing noise in the algorithm(s), and many more [3]. Many researchers are predicting that 10-15 years are needed for the widespread deployment of quantum computers [3, 12]. Based on this compliance, this article realizes a low-complexity hardware accelerator of present public-key algorithms over large key sizes for area-constrained cryptographic applications.  
As compared to RSA, ECC ensures equivalent security with shorter key lengths. For example, 233 bits of ECC are required to achieve identical security to a 2048-bit RSA [2]. The shorter key length assures additional benefits such as low area, low power consumption, a lower channel bandwidth requirement for data transmitting over the unsecured channel, etc. Therefore, ECC is an appropriate public-key cryptographic algorithm for area-constraint devices and applications. The NIST has standardized elliptic curve parameters for prime, i.e., \(GF(P)\), and binary, i.e., \(GF(2^{m})\), fields [13]. The NIST-specified lengths aredifferent for the prime field (192, 224, 256, 384, and 521) and binary field (163, 233, 283, 409 and 571). Furthermore, the prime field has been frequently targeted for software-based implementations [14]. On the other hand, the binary field is preferred for field-programmable gate arrays (FPGA) and application-specific integrated circuits (ASIC) [15; 16; 17; 18; 19; 20; 21; 22; 23; 24; 25].  
In addition to key length and the selection of a particular field, the choice between polynomial and normal basis representations is also important. These representations are required to implement the finite field modular operations (addition, multiplication, square, and inversion). The polynomial basis provides efficient modular multiplications while the normal basis is more suitable for modular square implementations [14; 26]. Furthermore, ECC offers two coordinate representations (affine and projective) to execute the required cryptographic operations. In comparison, the affine coordinate representation is more expensive in throughput as it needs an inversion operation [14; 17; 21; 25]. Therefore, we have preferred projective coordinates to implement \(GF(2^{409})\) and \(GF(2^{571})\) fields with polynomial basis representations for our computations in this work. The reason behind the selection of large field lengths is to consider higher security as the higher key lengths specify the higher security.