In this work we present the $łambda$-coordinates, a new system for representing points in binary elliptic curves. We also provide efficient elliptic curve operations based on the new representation and timing results of our software implementation over the field $픽_2^254$. As a result, we improve speed records for protected/unprotected single/multi-core software implementations of random-point elliptic curve scalar multiplication at the 128-bit security level. When implemented on a Sandy Bridge 3.4GHz Intel Xeon processor, our software is able to compute a single/multi-core unprotected scalar multiplication in 72,300 and 47,900 clock cycles, respectively; and a protected single-core scalar multiplication in 114,800 cycles. These numbers improve by around 2% on the newer Core i7 2.8GHz Ivy Bridge platform.