Solving 114-bit ECDLP for a Barreto-Naehrig Curve

Takuya Kusaka (1) , Sho Joichi (1) , Ken Ikuta (1) , Md Al-Amin Khandaker (1) , Yasuyuki Nogami (1) , Satoshi Uehara (2) , Nariyoshi Yamai (3) , Sylvain Duquesne (4)

1 Okayama University
2 UKK - University of Kitakyushu
3 TUAT - Tokyo University of Agriculture and Technology
4 IRMAR - Institut de Recherche Mathématique de Rennes

The security of cryptographic protocols which are based on elliptic curve cryptography relies on the intractability of elliptic curve discrete logarithm problem (ECDLP). In this paper, the authors describe techniques applied to solve 114-bit ECDLP in Barreto-Naehrig (BN) curve defined over the odd characteristic field. Unlike generic el-liptic curves, BN curve holds an especial interest since it is well studied in pairing-based cryptography. Till the date of our knowledge, the previous record for solving ECDLP in a prime field was 112-bit by Bos et al in Certicom curve 'secp112r1'. This work sets a new record by solving 114-bit prime field ECDLP of BN curve using Pollard's rho method. The authors utilized sextic twist property of the BN curve to efficiently carry out the random walk of Pollard's rho method. The parallel implementation of the rho method by adopting a client-server model, using 2000 CPU cores took about 6 months to solve the ECDLP.

Mots clés: Pollard's rho method; Barreto-Naehrig curve; ECDLP

Domaines: Théorie des nombres [math.NT]; Théorie de l'information et codage [math.IT]; Cryptographie et sécurité [cs.CR]

Takuya Kusaka, Sho Joichi, Ken Ikuta, Md Al-Amin Khandaker, Yasuyuki Nogami, et al.. Solving 114-bit ECDLP for a Barreto-Naehrig Curve. Information Security and Cryptology - ICISC 2017, Nov 2017, Séoul, South Korea. pp.231-244, ⟨10.1007/978-3-319-78556-1_13⟩. ⟨hal-01633653⟩