## Group-based CryptographyThis 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. |

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### Contents

I | 1 |

II | 3 |

III | 4 |

IV | 5 |

V | 6 |

VI | 7 |

VII | 9 |

VIII | 11 |

LVIII | 71 |

LIX | 72 |

LX | 77 |

LXI | 78 |

LXIII | 81 |

LXIV | 82 |

LXV | 84 |

LXVI | 87 |

XI | 12 |

XIII | 13 |

XIV | 14 |

XVI | 16 |

XVII | 17 |

XVIII | 19 |

XIX | 20 |

XX | 21 |

XXIII | 23 |

XXIV | 25 |

XXV | 26 |

XXVI | 27 |

XXVIII | 28 |

XXIX | 29 |

XXX | 30 |

XXXI | 31 |

XXXII | 33 |

XXXIII | 35 |

XXXIV | 37 |

XXXV | 39 |

XXXVI | 40 |

XXXVII | 41 |

XXXVIII | 42 |

XXXIX | 43 |

XLI | 45 |

XLII | 47 |

XLIII | 49 |

XLV | 50 |

XLVI | 51 |

XLVII | 52 |

XLVIII | 55 |

XLIX | 56 |

L | 59 |

LI | 60 |

LII | 61 |

LIII | 65 |

LIV | 67 |

LVI | 68 |

LXVII | 88 |

LXVIII | 89 |

LXIX | 94 |

LXX | 99 |

LXXI | 101 |

LXXII | 102 |

LXXIII | 103 |

LXXV | 109 |

LXXVII | 110 |

LXXVIII | 111 |

LXXIX | 114 |

LXXX | 116 |

LXXXI | 118 |

LXXXII | 120 |

LXXXIII | 121 |

LXXXIV | 122 |

LXXXV | 123 |

LXXXVII | 124 |

LXXXVIII | 125 |

LXXXIX | 129 |

XC | 131 |

XCII | 135 |

XCIII | 139 |

XCIV | 141 |

XCV | 142 |

XCVI | 143 |

XCVII | 145 |

XCVIII | 148 |

XCIX | 150 |

C | 152 |

CII | 155 |

CIII | 156 |

CIV | 157 |

CV | 161 |

CVI | 162 |

CVII | 164 |

CVIII | 167 |

### Common terms and phrases

ˆΓ abelian adversary algorithmic problems Alice Alice and Bob Alice’s asymptotic density average case complexity Bob’s braid groups braid word combinatorial group theory compute the geodesic conﬁguration conjugacy search problem cryptographic protocol decision algorithm decision problem decomposition search problem decryption deﬁning relators Deﬁnition denote diﬀerent distribution eﬃcient elements encryption ensemble equations example exponentially ﬁnd ﬁnite ﬁnite generating set ﬁnite set ﬁnitely generated subgroup ﬁnitely presented groups ﬁrst ﬁxed free basis property free group function f geodesic length given group G inﬁnite isomorphism Lemma Let G linear many-one reductions metabelian groups natural number nonabelian nonabelian groups normal form platform group polynomial polynomial on average probability Proof public key cryptography quasi-isometrically embedded randomly reduced satisﬁes SCSP secret key Section set of inputs solution solvable solve speciﬁc spherical stratiﬁcation subgroup of G subset suﬃcient Theorem tuple Turing machine word problem