Integer and polynomial multiplication: towards optimal toom-cook matrices
Proceedings of the 2007 international symposium on Symbolic and algebraic computation
High-speed software implementation of the optimal ate pairing over Barreto-Naehrig curves
Pairing'10 Proceedings of the 4th international conference on Pairing-based cryptography
A family of implementation-friendly BN elliptic curves
Journal of Systems and Software
Faster explicit formulas for computing pairings over ordinary curves
EUROCRYPT'11 Proceedings of the 30th Annual international conference on Theory and applications of cryptographic techniques: advances in cryptology
Faster squaring in the cyclotomic subgroup of sixth degree extensions
PKC'10 Proceedings of the 13th international conference on Practice and Theory in Public Key Cryptography
Finding optimal formulae for bilinear maps
WAIFI'12 Proceedings of the 4th international conference on Arithmetic of Finite Fields
NEON implementation of an attribute-based encryption scheme
ACNS'13 Proceedings of the 11th international conference on Applied Cryptography and Network Security
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We present efficient squaring formulae based on the Toom-Cook multiplication algorithm. The latter always requires at least one non-trivial constant division in the interpolation step. We show such non-trivial divisions are not needed in the case two operands are equal for three, four and five-way squarings. Our analysis shows that our 3-way squaring algorithms have much less overhead than the best known 3-way Toom-Cook algorithm. Our experimental results show that one of our new 3-way squaring methods performs faster than mpz_mul() in GNU multiple precision library (GMP) for squaring integers of approximately 2400-6700 bits on Pentium IV Prescott 3.2GHz. For squaring in Z[x], our 3-way squaring algorithms are much superior to other known squaring algorithms for small input size. In addition, we present 4-way and 5-way squaring formulae which do not require any constant divisions by integers other than a power of 2. Under some reasonable assumptions, our 5-way squaring formula is faster than the recently proposed Montgomery's 5-way Karatsuba-like formulae.