Fast ECDH Key Exchange Using Twisted Edwards Curves with an Efficiently Computable Endomorphism

It is widely accepted that public-key cryptosystems play a major role in the security arena of the Internet of Things (IoT), but they need to be implemented efficiently to not deplete the scarce resources of battery-operated devices such as wireless sensor nodes. This paper describes a highly-optimized software implementation of scalar multiplication for Elliptic Curve Diffie-Hellman (ECDH) key exchange on resource-limited IoT devices that achieves fast execution times along with reasonably small code size and RAM consumption. Our software uses a special class of elliptic curves, namely twisted Edwards curves with an efficiently computable endomorphism similar to that of the socalled Gallant-Lambert-Vanstone (GLV) curves. This allows us to combine the main advantage of the GLV model, which is an efficiently-computable endomorphism to speed up variable-base scalar multiplication, with the fast and complete addition rules of the (twisted) Edwards model. We implemented variable-base scalar multiplication for static ECDH on two such curves, one over a 159-bit and the second over a 207-bit pseudo-Mersenne prime field, respectively, and evaluated their execution time on a 16-bit MSP430F1611 processor. The arithmetic operations in the prime field do not contain operand-dependent conditional statements (in particular no “if-then-else” clauses) and also the scalar multiplication follows a fixed execution path for a given (static) scalar. A variable-base scalar multiplication on curves over the 159 and 207-bit field takes about 2.63 and 4.84 million clock cycles, respectively, on an MSP430F1611 processor. These results compare favorably with the Montgomery ladder on the equivalent Montgomery curves, which is almost 50% slower.