A public key cryptosystem and a signature scheme based on discrete logarithms
Proceedings of CRYPTO 84 on Advances in cryptology
Discrete logarithms in finite fields and their cryptographic significance
Proc. of the EUROCRYPT 84 workshop on Advances in cryptology: theory and application of cryptographic techniques
Handbook of Applied Cryptography
Handbook of Applied Cryptography
ACISP '97 Proceedings of the Second Australasian Conference on Information Security and Privacy
SAC '01 Revised Papers from the 8th Annual International Workshop on Selected Areas in Cryptography
CRYPTO '00 Proceedings of the 20th Annual International Cryptology Conference on Advances in Cryptology
Solving Large Sparse Linear Systems over Finite Fields
CRYPTO '90 Proceedings of the 10th Annual International Cryptology Conference on Advances in Cryptology
ASIACRYPT '99 Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology
ASIACRYPT '94 Proceedings of the 4th International Conference on the Theory and Applications of Cryptology: Advances in Cryptology
A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
A theoretical basis for the reduction of polynomials to canonical forms
ACM SIGSAM Bulletin
Practical cryptography in high dimensional tori
EUROCRYPT'05 Proceedings of the 24th annual international conference on Theory and Applications of Cryptographic Techniques
Applications to cryptography of twisting commutative algebraic groups
Discrete Applied Mathematics
Journal of Symbolic Computation
On Cryptographic Schemes Based on Discrete Logarithms and Factoring
CANS '09 Proceedings of the 8th International Conference on Cryptology and Network Security
The number field sieve in the medium prime case
CRYPTO'06 Proceedings of the 26th annual international conference on Advances in Cryptology
The function field sieve in the medium prime case
EUROCRYPT'06 Proceedings of the 24th annual international conference on The Theory and Applications of Cryptographic Techniques
Pairing-Based cryptography at high security levels
IMA'05 Proceedings of the 10th international conference on Cryptography and Coding
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Using a recent idea of Gaudry and exploiting rational representations of algebraic tori, we present an index calculus type algorithm for solving the discrete logarithm problem that works directly in these groups. Using a prototype implementation, we obtain practical upper bounds for the difficulty of solving the DLP in the tori $T_2(\mathbb{F}_{p^m})$ and $T_6(\mathbb{F}_{p^m})$ for various p and m. Our results do not affect the security of the cryptosystems LUC, XTR, or CEILIDH over prime fields. However, the practical efficiency of our method against other methods needs further examining, for certain choices of p and m in regions of cryptographic interest.