Recent algorithmic improvements of discrete logarithm computation in special extension fields threaten the security of pairing-friendly curves used in practice. A possible answer to this delicate situation is to propose alternative curves that are immune to these attacks, without
compromising the efficiency of the pairing computation too much. We follow this direction, and focus on embedding degrees 5 to 8; we extend the Cocks–Pinch algorithm to obtain pairing-friendly curves with an efficient ate pairing. We carefully select our curve parameters so as to thwart possible attacks by « special » or « tower » Number Field Sieve algorithms. We target a 128-bit security level, and back this security claim by computation time estimates for the DLP computation. We also compare the efficiency of the ate pairing computation on these curves to k = 12 curves (Barreto–Naehrig, Barreto–Lynn–Scott), k = 16 curves (Kachisa–Schaefer–Scott) and k = 1 curves (Chatterjee–Menezes–Rodríguez-Henríquez).