On arithmetical algorithms over finite fields
Journal of Combinatorial Theory Series A
On fast multiplication of polynomials over arbitrary algebras
Acta Informatica
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The Design and Analysis of Computer Algorithms
The Design and Analysis of Computer Algorithms
More on Squaring and Multiplying Large Integers
IEEE Transactions on Computers
Low-Weight Polynomial Form Integers for Efficient Modular Multiplication
IEEE Transactions on Computers
A New Approach to Subquadratic Space Complexity Parallel Multipliers for Extended Binary Fields
IEEE Transactions on Computers
Comments on "Five, Six, and Seven-Term Karatsuba-Like Formulae"
IEEE Transactions on Computers
A Novel Architecture for Galois Fields GF(2^m) Multipliers Based on Mastrovito Scheme
IEEE Transactions on Computers
On multiplication in finite fields
Journal of Complexity
Explicit formulas for efficient multiplication in F36m
SAC'07 Proceedings of the 14th international conference on Selected areas in cryptography
Efficient multiplication in F3lm, m ≥ 1 and 5 ≤ l ≤ 18
AFRICACRYPT'08 Proceedings of the Cryptology in Africa 1st international conference on Progress in cryptology
Faster multiplication in GF(2)[x]
ANTS-VIII'08 Proceedings of the 8th international conference on Algorithmic number theory
Proceedings of the ACM SIGCOMM 2010 conference
IEEE Transactions on Very Large Scale Integration (VLSI) Systems
Delaying mismatched field multiplications in pairing computations
WAIFI'10 Proceedings of the Third international conference on Arithmetic of finite fields
An analysis of affine coordinates for pairing computation
Pairing'10 Proceedings of the 4th international conference on Pairing-based cryptography
Compact hardware for computing the tate pairing over 128-bit-security supersingular curves
Pairing'10 Proceedings of the 4th international conference on Pairing-based cryptography
Multiplication of polynomials modulo xn
Theoretical Computer Science
Efficient multiplication in finite field extensions of degree 5
AFRICACRYPT'11 Proceedings of the 4th international conference on Progress in cryptology in Africa
CHES'11 Proceedings of the 13th international conference on Cryptographic hardware and embedded systems
Impact of Intel's new instruction sets on software implementation of GF(2)[x] multiplication
Information Processing Letters
Parallelizing the weil and tate pairings
IMACC'11 Proceedings of the 13th IMA international conference on Cryptography and Coding
Efficient multiplication over extension fields
WAIFI'12 Proceedings of the 4th international conference on Arithmetic of Finite Fields
Finding optimal formulae for bilinear maps
WAIFI'12 Proceedings of the 4th international conference on Arithmetic of Finite Fields
WEWoRC'11 Proceedings of the 4th Western European conference on Research in Cryptology
Faster implementation of scalar multiplication on koblitz curves
LATINCRYPT'12 Proceedings of the 2nd international conference on Cryptology and Information Security in Latin America
The M4RIE library for dense linear algebra over small fields with even characteristic
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
New bit parallel multiplier with low space complexity for all irreducible trinomials over GF(2n)
IEEE Transactions on Very Large Scale Integration (VLSI) Systems
Information Processing Letters
| Hi-index | 14.99 |
The Karatsuba-Ofman algorithm starts with a way to multiply two 2-term (i.e., linear) polynomials using three scalar multiplications. There is also a way to multiply two 3-term (i.e., quadratic) polynomials using six scalar multiplications. These are used within recursive constructions to multiply two higher-degree polynomials in subquadratic time. We present division-free formulae which multiply two 5-term polynomials with 13 scalar multiplications, two 6-term polynomials with 17 scalar multiplications, and two 7-term polynomials with 22 scalar multiplications. These formulae may be mixed with the 2-term and 3-term formulae within recursive constructions, leading to improved bounds for many other degrees. Using only the 6-term formula leads to better asymptotic performance than standard Karatsuba. The new formulae work in any characteristic, but simplify in characteristic 2. We describe their application to elliptic curve arithmetic over binary fields. We include some timing data.