In today's technology, a sheer number of Internet of Things applications use hardware security modules for secure communications. The widely used algorithms in security modules, for example, digital signatures and key agreement, are based upon elliptic curve cryptography (ECC). A core operation used in ECC is the point multiplication, which is computationally expensive for many Internet of things applications. In many IoT applications, such as intelligent transportation systems and distributed control systems, thousands of safety messages need to be signed and verified within a very short time-frame. Considerable research has been conducted in the design of a fast elliptic curve arithmetic on finite fields using residue number systems (RNS). In this article, we propose an RNS-based ECC core hardware for the two families of elliptic curves that are short Weierstrass and twisted Edwards curves. Specifically, we present RNS implementations for SECP256K1 and ED25519 standard curves. We propose an RNS hardware architecture supporting fast elliptic curve point-addition (ECPA), point-doubling (ECPD), and point-tripling (ECPT). We implemented different ECC point multiplication algorithms on the Xilinx FPGA platform. The test results confirm that the performance of our fully RNS ECC point multiplication is better than the fastest ECC point multiplication cores in the literature.

In this study, the authors propose a new protocol to share secret shadows for verifiable secret sharing (VSS) schemes. Unlike traditional VSS schemes, whose communications between the dealer and the participants require a secure channel, the authors’ new scheme relies on the elliptic curve cryptosystem and the Chinese remainder theorem operates over a public channel. The security of the secret shadows and the verification algorithm are based on the hardness of the elliptic curve discrete logarithm problem. They also extend the proposed scheme to an efficient verifiable multi‐secret sharing (VMSS) scheme, particularly when the number of secrets is more than the threshold. As a result, their scheme is a multi‐use and efficient VMSS on the public channel which provides the same level of security as traditional VMSS schemes with much shorter keys.

We address several algorithms to perform a double-scalar multiplication on an elliptic curve. All the methods investigated are related to the double-base number system (DBNS) and extend previous work of Doche et al. [25]. We refine and rigorously prove the complexity analysis of the joint binary-ternary (JBT) algorithm. Experiments are in line with the theory and show that the JBT requires approximately 6 percent less field multiplications than the standard joint sparse form (JSF) method to compute [n]P + [m]Q. We also introduce a randomized version of the JBT, called JBT-Rand, that gives total control of the number of triplings in the expansion that is produced. So it becomes possible with the JBT-Rand to adapt and tune the number of triplings to the coordinate system and bit length that are used, to further decrease the cost of a double-scalar multiplication. Then, we focus on Koblitz curves. For extension degrees enjoying an optimal normal basis of type II, we discuss a Joint τ-DBNS approach that reduces the number of field multiplications by at least 35 percent over the traditional τ-JSF. For other extension degrees represented in polynomial basis, the Joint τ-DBNS is still relevant provided that appropriate bases conversion methods are used. In this situation, tests show that the speedup over the τ-JSF is then larger than 20 percent. Finally, when the use of the τ-DBNS becomes unrealistic, for instance because of the lack of an efficient normal basis or the lack of memory to allow an efficient conversion, we adapt the joint binary-ternary algorithm to Koblitz curves giving rise to the Joint τ-τ method whose complexity is analyzed and proved. The Joint τ-τ induces a speedup of about 10 percent over the τ-JSF.