Open AccessJournal Article
Torus-based cryptography
Karl Rubin,Alice Silverberg +1 more
91
TL;DR: In this paper, the authors introduce the concept of torus-based cryptography, give a new public key system called CEILIDH, and compare it to other discrete log based systems including Lucas-based systems and XTR.
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Abstract: We introduce the concept of torus-based cryptography, give a new public key system called CEILIDH, and compare it to other discrete log based systems including Lucas-based systems and XTR. Like those systems, we obtain small key sizes. While Lucas-based systems and XTR are essentially restricted to exponentiation, we are able to perform multiplication as well. We also disprove the open conjectures from [2], and give a new algebro-geometric interpretation of the approach in that paper and of LUC and XTR.
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Citations
VSH, an efficient and provable collision-resistant hash function
Scott Contini,Arjen K. Lenstra,Ron Steinfeld +2 more
- 28 May 2006
TL;DR: VSH, very smooth hash, a new S-bit hash function that is provably collision-resistant assuming the hardness of finding nontrivial modular square roots of very smooth numbers modulo an S- bit composite, is introduced.
•Journal Article
The function field sieve in the medium prime case
Antoine Joux,Reynald Lercier +1 more
TL;DR: This approach is an alternative to a recent paper of Granger and Vercauteren for computing discrete logarithms in tori, using efficient torus representations, and shows that when q is not too large, a very efficient L(1/3) variation of the function field sieve can be used.
101
•Posted Content
On Small Characteristic Algebraic Tori in Pairing-Based Cryptography.
TL;DR: This work transfers techniques recently developed for torus-based cryptography to pairing- based cryptography, resulting in more efficient computations, and lower bandwidth requirements, to illustrate the efficacy of this approach.
A comparison of CEILIDH and XTR
Robert Granger,Daniel Page,Martijn Stam +2 more
- 13 Jun 2004
TL;DR: An efficient version of CEILIDH is developed and it is shown that while it seems bound to be inherently slower than XTR, the difference in performance is much smaller than what one might infer from the original description.
52
On the discrete logarithm problem on algebraic tori
Robert Granger,Frederik Vercauteren +1 more
- 14 Aug 2005
TL;DR: An index calculus type algorithm for solving the discrete logarithm problem that works directly in algebraic tori groups and practical upper bounds for the difficulty of solving the DLP in the tori are obtained.
References
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Handbook of Applied Cryptography
Alfred Menezes,Scott A. Vanstone,Paul C. van Oorschot +2 more
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TL;DR: A valuable reference for the novice as well as for the expert who needs a wider scope of coverage within the area of cryptography, this book provides easy and rapid access of information and includes more than 200 algorithms and protocols.
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Adeles and algebraic groups
André Weil
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TL;DR: Ono and Ono as mentioned in this paper gave a short survey of subsequent research on Adele-Geometry and the Tamagawa numbers, including the case of the local measure and the convergence factor.
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Algebraic Groups and Their Birational Invariants
V. Voskresenskiĭ
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TL;DR: In this paper, the main topics are forms and Galois cohomology, the Picard group and the Brauer group, birational geometry of algebraic tori, algebraic groups, Tamagawa numbers, $R$-equivalence, projective toric varieties, invariants of finite transformation groups, and index-formulas.
336
The XTR Public Key System
Arjen K. Lenstra,Eric R. Verheul +1 more
- 20 Aug 2000
TL;DR: The XTR public key system as discussed by the authors is based on a new method to represent elements of a subgroup of a multiplicative group of a finite field, which leads to substantial savings both in communication and computational overhead without compromising security.
Arithmetic of Algebraic Tori
TL;DR: In this article, the authors present a structure theorem for C(CJA,Ir, (T)k) (resp. c()A, r)) 3.1.
187
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