Group-based CryptographySpringer Science & Business Media, 2008年7月17日 - 183 頁 This book is about relations between three di?erent areas of mathematics and theoreticalcomputer science: combinatorialgroup theory, cryptography, and c- plexity theory. We explorehownon-commutative(in?nite) groups, which arety- callystudiedincombinatorialgrouptheory, canbeusedinpublickeycryptography. We also show that there is a remarkable feedback from cryptography to com- natorial group theory because some of the problems motivated by cryptography appear to be new to group theory, and they open many interesting research - enues within group theory. Then, we employ complexity theory, notably generic case complexity of algorithms, for cryptanalysisof various cryptographicprotocols based on in?nite groups. We also use the ideas and machinery from the theory of generic case complexity to study asymptotically dominant properties of some in?nite groups that have been used in public key cryptography so far. It turns out that for a relevant cryptographic scheme to be secure, it is essential that keys are selected from a "very small" (relative to the whole group, say) subset rather than from the whole group. Detecting these subsets ("black holes") for a part- ular cryptographic scheme is usually a very challenging problem, but it holds the keyto creatingsecurecryptographicprimitives basedonin?nite non-commutative groups. The book isbased onlecture notesfor the Advanced Courseon Group-Based CryptographyheldattheCRM, BarcelonainMay2007. Itisagreatpleasureforus to thank Manuel Castellet, the HonoraryDirector of the CRM, for supporting the idea of this Advanced Course. We are also grateful to the current CRM Director, JoaquimBruna, and to the friendly CRM sta?, especially Mrs. N. PortetandMrs. N. Hern ́ andez, for their help in running the Advanced Course and in preparing the lecture notes. |
內容
Background on Public Key Cryptography | 3 |
2 | 12 |
Background on Computational Complexity | 19 |
3 | 28 |
3Satisfiability | 29 |
Canonical Noncommutative Cryptography | 37 |
1 | 55 |
5 | 68 |
12 | 81 |
Distributional Problems and the Average Case Complexity | 99 |
13 | 109 |
Generic Case Complexity 117 | 115 |
The linear generic time complexity of subset sum problem | 129 |
Asymptotically Dominant Properties | 139 |
LengthBased and Quotient Attacks | 148 |
Bibliography | 169 |
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常見字詞
abelian algebraic algorithmic problems Alice and Bob Artin groups asymptotic density automorphism average case complexity average functions behavior Bob’s braid groups braid word commuting subgroups conjugacy problem conjugacy search problem correctly with probability cryptographic protocol decision problems decomposition problem decomposition search problem defining relators definition denote distributions ensemble equal equations example exists exponentially free group function f generic-case geodesic length given group G group theory infinite integers invertible isomorphism Lemma Let G linear many-one reductions matrices metabelian groups nonabelian normal form pair platform group polynomial on average private key Proof protocol based public key public key cryptography quotient attacks r₁ random word randomly recursive reduced satisfying scheme SCSP secret key Section seminormal seminormal form sequence Small cancellation groups solving spheres spherical Stickel's subgroup of G subgroup-restricted subset subword Theorem Tietze transformations transmitted trivial Turing machine w₁