Structure of parallel multipliers for a class of fields GF(2m)
Information and Computation
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
The MAGMA algebra system I: the user language
Journal of Symbolic Computation - Special issue on computational algebra and number theory: proceedings of the first MAGMA conference
A Fast Algorithm for Multiplicative Inversion in GF(2m) Using Normal Basis
IEEE Transactions on Computers
Polynomial-time quantum algorithms for Pell's equation and the principal ideal problem
STOC '02 Proceedings of the thiry-fourth annual ACM symposium on Theory of computing
Quantum computation and quantum information
Quantum computation and quantum information
Quantum algorithms for some hidden shift problems
SODA '03 Proceedings of the fourteenth annual ACM-SIAM symposium on Discrete algorithms
VLSI Designs for Multiplication over Finite Fields GF (2m)
AAECC-6 Proceedings of the 6th International Conference, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Optimal Extension Fields for Fast Arithmetic in Public-Key Algorithms
CRYPTO '98 Proceedings of the 18th Annual International Cryptology Conference on Advances in Cryptology
Redundant Representation of Finite Fields
PKC '01 Proceedings of the 4th International Workshop on Practice and Theory in Public Key Cryptography: Public Key Cryptography
Fast Multiplication in Finite Fields GF(2N)
CHES '99 Proceedings of the First International Workshop on Cryptographic Hardware and Embedded Systems
A Fast Implementation of Multiplicative Inversion Over GF(2^m )
ITCC '05 Proceedings of the International Conference on Information Technology: Coding and Computing (ITCC'05) - Volume I - Volume 01
Software Multiplication Using Gaussian Normal Bases
IEEE Transactions on Computers
The Power of Strong Fourier Sampling: Quantum Algorithms for Affine Groups and Hidden Shifts
SIAM Journal on Computing
Quantum Algorithms for Hidden Nonlinear Structures
FOCS '07 Proceedings of the 48th Annual IEEE Symposium on Foundations of Computer Science
Quantum algorithms for highly non-linear Boolean functions
SODA '10 Proceedings of the twenty-first annual ACM-SIAM symposium on Discrete Algorithms
An O(m2)-depth quantum algorithm for the elliptic curve discrete logarithm problem over GF(2m)a
Quantum Information & Computation
Quantum universality by state distillation
Quantum Information & Computation
Proceedings of the 3rd Innovations in Theoretical Computer Science Conference
Low complexity bit parallel architectures for polynomial basis multiplication over GF(2m)
IEEE Transactions on Computers
Efficient quantum circuits for binary elliptic curve arithmetic: reducing T-gate complexity
Quantum Information & Computation
Hi-index | 0.00 |
Finite fields of the form F2m play an important role in coding theory and cryptography. We show that the choice of how to represent the elements of these fields can have a significant impact on the resource requirements for quantum arithmetic. In particular, we show how the use of Gaussian normal basis representations and of 'ghost-bit basis' representations can be used to implement inverters with a quantum circuit of depth O(mlog(m)). To the best of our knowledge, this is the first construction with sub-quadratic depth reported in the literature. Our quantum circuit for the computation of multiplicative inverses is based on the Itoh-Tsujii algorithm which exploits that in normal basis representation squaring corresponds to a permutation of the coefficients. We give resource estimates for the resulting quantum circuit for inversion over binary fields F2m based on an elementary gate set that is useful for fault-tolerant implementation.