New security issues and designs for the Internet of Things (IoT) have been an important area of research due to the emergence of IoT applications. In general, the use cases include resource constrained embedded processors, which introduce challenges on the use of Public Key Cryptography (PKC) due to the lack of available memory and higher costs attached to energy consumption. PKC protocols including digital signatures and key agreement schemes usually lead to significant overheads in terms of execution time and energy consumption, which are undesirable for the low-end batterypowered IoT processors. A common sensor node features an 8- bit microcontroller clocked at a frequency of less than 32 MHz and equipped with a few kilobytes (KB) of data and stack memories and up to 256 KB of flash memory for storing program code and constants. Under these computational constraints, the most precious resource of IoT device is energy. Once deployed in the field, the platforms are expected to work several months or years, with the limited energy supplied by two AA batteries that cannot be easily replaced or even recharged. To minimize the energy consumption, lightweight PKC implementations are a fundamental requirement. Among the existing candidates for PKC, elliptic curve cryptography (ECC) is the most promising one compared to other finite field-based (Diffie-Hellman) or integer-factoring-based cryptosystems (RSA). While the best classical attacks against RSA and Diffie-Hellman over finite fields are sub-exponential attacks while the Elliptic Curve Discrete Logarithm Problem (ECDLP) remains exponential and are based on the Pollard’s rho attack. This translates into smaller key sizes and bandwidth occupation for ECC. The features of ECC has also promote the public-key cryptography application in IoT solutions as, in many cases, transmitting a bit is about orders of magnitude more expensive than processing a bit in terms of energy consumption in Wireless Sensor Networks for example. Many elliptic curves are already standardized for several years, for example, NIST standardized 15 elliptic curves, designed by NSA, in the FIPS 186-2 standard together with the ECDSA digital signature scheme in the year of 2000. On the other hand, Edward Snowden, who worked many years for NSA revealed in 2013 that the Dual EC DRBG algorithm, presented as a cryptographically secure pseudorandom number generator (CSPRNG), would possibly contain a backdoor. In addition to that, new elliptic curves models and optimized parameters with protection against side-channel attacks have been found in the last years and therefore there is consensus in the cryptographic community that elliptic curve standards should be upgraded.