Use of elliptic curves in cryptography
Lecture notes in computer sciences; 218 on Advances in cryptology---CRYPTO 85
A hierarchy of polynomial time lattice basis reduction algorithms
Theoretical Computer Science
Lattice basis reduction: improved practical algorithms and solving subset sum problems
Mathematical Programming: Series A and B
Identity-Based Encryption from the Weil Pairing
CRYPTO '01 Proceedings of the 21st Annual International Cryptology Conference on Advances in Cryptology
Low Complexity Multiplication in a Finite Field Using Ring Representation
IEEE Transactions on Computers
Modern Computer Algebra
Short Signatures from the Weil Pairing
Journal of Cryptology
Five, Six, and Seven-Term Karatsuba-Like Formulae
IEEE Transactions on Computers
WAIFI '07 Proceedings of the 1st international workshop on Arithmetic of Finite Fields
Efficient Modular Arithmetic in Adapted Modular Number System Using Lagrange Representation
ACISP '08 Proceedings of the 13th Australasian conference on Information Security and Privacy
Finite Field Multiplication Combining AMNS and DFT Approach for Pairing Cryptography
ACISP '09 Proceedings of the 14th Australasian Conference on Information Security and Privacy
Fast point multiplication on elliptic curves through isogenies
AAECC'03 Proceedings of the 15th international conference on Applied algebra, algebraic algorithms and error-correcting codes
EUROCRYPT'08 Proceedings of the theory and applications of cryptographic techniques 27th annual international conference on Advances in cryptology
Implementing Gentry's fully-homomorphic encryption scheme
EUROCRYPT'11 Proceedings of the 30th Annual international conference on Theory and applications of cryptographic techniques: advances in cryptology
Efficient multiplication in finite field extensions of degree 5
AFRICACRYPT'11 Proceedings of the 4th international conference on Progress in cryptology in Africa
Modular number systems: beyond the mersenne family
SAC'04 Proceedings of the 11th international conference on Selected Areas in Cryptography
Practical cryptography in high dimensional tori
EUROCRYPT'05 Proceedings of the 24th annual international conference on Theory and Applications of Cryptographic Techniques
Lattice enumeration using extreme pruning
EUROCRYPT'10 Proceedings of the 29th Annual international conference on Theory and Applications of Cryptographic Techniques
Rings of Low Multiplicative Complexity
Finite Fields and Their Applications
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The efficiency of cryptographic protocols rely on the speed of the underlying arithmetic and finite field computation. In the literature, several methods on how to improve the multiplication over extensions fields $\mathbb{F}_{q^{m}}$, for prime q were developped. These optimisations are often related to the Karatsuba and Toom Cook methods. However, the speeding-up is only interesting when m is a product of powers of 2 and 3. In general cases, a fast multiplication over $\mathbb{F}_{q^{m}}$ is implemented through the use of the naive school-book method. In this paper, we propose a new efficient multiplication over $\mathbb{F}_{q^{m}}$ for any power m. The multiplication relies on the notion of Adapted Modular Number System (AMNS), introduced in 2004 by [3]. We improve the construction of an AMNS basis and we provide a fast implementation of the multiplication over $\mathbb{F}_{q^{m}}$, which is faster than GMP and NTL.