The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
The art of computer programming, volume 2 (3rd ed.): seminumerical algorithms
A survey of fast exponentiation methods
Journal of Algorithms
Normal bases via general Gauss periods
Mathematics of Computation
On the Generalised Hidden Number Problem and Bit Security of XTR
AAECC-14 Proceedings of the 14th International Symposium on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
Gauss Periods and Fast Exponentiation in Finite Fields (Extended Abstract)
LATIN '95 Proceedings of the Second Latin American Symposium on Theoretical Informatics
SAC '01 Revised Papers from the 8th Annual International Workshop on Selected Areas in Cryptography
Optimal Extension Fields for XTR
SAC '02 Revised Papers from the 9th Annual International Workshop on Selected Areas in Cryptography
CRYPTO '00 Proceedings of the 20th Annual International Cryptology Conference on Advances in Cryptology
Hidden Number Problem with the Trace and Bit Security of XTR and LUC
CRYPTO '02 Proceedings of the 22nd Annual International Cryptology Conference on Advances in Cryptology
Evidence that XTR Is More Secure than Supersingular Elliptic Curve Cryptosystems
EUROCRYPT '01 Proceedings of the International Conference on the Theory and Application of Cryptographic Techniques: Advances in Cryptology
Fast Generation of Cubic Irreducible Polynomials for XTR
INDOCRYPT '01 Proceedings of the Second International Conference on Cryptology in India: Progress in Cryptology
ASIACRYPT '99 Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology
ASIACRYPT '00 Proceedings of the 6th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
ASIACRYPT '01 Proceedings of the 7th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
ASIACRYPT '02 Proceedings of the 8th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
Fast Irreducibility and Subgroup Membership Testing in XTR
PKC '01 Proceedings of the 4th International Workshop on Practice and Theory in Public Key Cryptography: Public Key Cryptography
Efficient Subgroup Exponentiation in Quadratic and Sixth Degree Extensions
CHES '02 Revised Papers from the 4th International Workshop on Cryptographic Hardware and Embedded Systems
The GH Public-Key Cryptosystem
SAC '01 Revised Papers from the 8th Annual International Workshop on Selected Areas in Cryptography
Public-key cryptosystems based on cubic finite field extensions
IEEE Transactions on Information Theory
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XTR appeared in 2000 is a very promising alternative to elliptic curve cryptosystem. Though the basic idea behind XTR is very elegant and universal, one needs to restrict the primes p such as p ≡ 2 mod3 for optimal normal bases since it involves many multiplications in GF(p2). Moreover the restriction p ≡ 2 mod3 is consistently used to improve the time complexity for irreducibility testing for XTR polynomials. In this paper, we propose that a Gaussian normal basis of type (2,k) for small k can also be used for efficient field arithmetic for XTR when p ≢ 2(mod 3). Furthermore we give a new algorithm for fast irreducibility testing and finding a generator of XTR group when p ≡ 1 mod 3. Also we present an explicit generator of XTR group which does not need any irreducibility testing when there is a Gaussian normal basis of type (2,3) in GF(p2). We show that our algorithms are simple to implement and the time complexity of our methods are comparable to the best ones proposed so far.