Improved algorithms for finding fixed-degree isogenies between supersingular elliptic curves

Finding isogenies between supersingular elliptic curves is a natural algorithmic problem which is known to be equivalent to computing the curves\u27 endomorphism rings. When the isogeny is additionally required to have a specific degree $d$, the problem appears to be somewhat different in nature, yet it is also considered a hard problem in isogeny-based cryptography. Let $E_1,E_2$ be supersingular elliptic curves over $\mathbb{F}_{p^2}$. We present improved classical and quantum algorithms that compute an isogeny of degree $d$ between $E_1$ and $E_2$ if it exists. Let the sought-after degree be $d = p^{1/2+ \epsilon}$ for some $\epsilon>0$. Our essentially memory-free algorithms have better time complexity than meet-in-the-middle algorithms, which require exponential memory storage, in the range $1/2\leq\epsilon\leq 3/4$ on a classical computer and quantum improvements in the range $0<\epsilon<5/2$

Topological groups

Namen tega diplomskega dela je predstaviti pojem topološke grupe in dokazati nekaj temeljnih izrekov iz študija topoloških grup. Definirana je topološka grupa in opisane so njene osnovne lastnosti. Obravnavane so topološke podgrupe in kvocientni topološki prostori topoloških grup. Pokazano je, da za topološke grupe veljajo podobni trije izreki o topoloških izomorfizmih kot za grupe. Na topološko grupo sta vpeljani leva in desna uniformna struktura, glede na kateri je vsaka topološka grupa uniformni prostor. Na topološki grupi je nato skonstruirana levoinvariantna psevdometrika. Karakterizirana je metrizabilnost za Hausdorffove topološke grupe in dokazano je, da sta za topološke grupe povsem regularnost in separacijski aksiom $T_0$ ekvivalentna. Skonstruiran je primer povsem regularne topološke grupe, ki ni normalna. Za regularne topološke prostore so navedene karakterizacije parakompaktnosti. Dokazano je, da je vsaka lokalno kompaktna Hausdorffova topološka grupa parakompaktna in posledično normalna.The goal of this thesis is to present the concept of a topological group and to prove some fundamental theorems from the study of topological groups. We define a topological group and describe its basic properties. We look at topological subgroups and quotient topological spaces of topological groups. We show that for topological groups three topological isomorphism theorems hold which are similar to those for groups. We introduce left and right uniform structures on a topological group and then show that every topological space is also a uniform space. We then construct a left invariant pseudo-metric on a topological group. We characterize metrizability for Hausdorff topological groups and we prove that complete regularity and the $T_0$ separation axiom are equivalent for topological groups. We construct an example of a completely regular topological group which is not a normal topological space. For regular topological spaces we list different characterizations of paracompactness. We then prove that every locally compact Hausdorff topological group is paracompact, and hence a normal topological space

Non-commutative Gröbner bases and improvements of Buchberger\u27s algorithm

Namen tega magistrskega dela je predstaviti teorijo Gröbnerjevih baz idealov v kolobarju nekomutativnih polinomov in tri glavne algoritme za njihov izračun, Buchbergerjev algoritem ter Faugèrjeva algoritma $F_4$ in $F_5$. Začnemo pri osnovah teorije nekomutativnih polinomov, predstavimo algoritem deljenja, definiramo Gröbnerjeve baze idealov nekomutativnih polinomov in dokažemo nekaj njihovih temeljnih lastnosti. Nadaljujemo s klasičnim Buchbergerjevim algoritmom, vpeljemo pojem ovire in nato sledimo korakom Tea More do nekomutativne različice algoritma. Pri tem z Dicksonovo lemo pokažemo končnost prvotnega komutativnega Buchbergerjevega algoritma ter dokažemo nekomutativno različico Buchbergerjevega kriterija in pravilnost nekomutativnega Buchbergerjevega algoritma. Pokažemo, kako množice polinomov pretvoriti v matrike ter hkrati formuliramo komutativen in nekomutativen algoritem $F_4$. Dokažemo pravilnost algoritma $F_4$ in pod pogojem, da za dani ideal obstaja končna Gröbnerjeva baza, dokažemo končnost algoritma F4. Definiramo modul vezi množice polinomov in dokažemo nekaj osnovnih lastnosti. Buchbergerjevo teorijo dvignemo v prosti modul nad kolobarjem komutativnih polinomov in definiramo polinomske podpise. Predstavimo osnovnega predstavnika družine podpisnih algoritmov ter dokažemo njegovo pravilnost in končnost. Vpeljemo kriterij $F_5$ in podpisni algoritem uporabimo, da formuliramo algoritem $F_5$. Za konec ponovimo prejšnje korake in predstavimo podpisni algoritem za nekomutativne polinome in dokažemo pravilnost nekomutativnega algoritma $F_5$. Pod pogojem, da za dani ideal obstaja končna Gröbnerjeva baza, dokažemo še njegovo končnost.The goal of this Master\u27s thesis is to present the theory of Gröbner bases of ideals in the ring of non-commutative polynomials, and the three main algorithms for computing them, Buchberger\u27s algorithm and Faugère\u27s $F_4$ and $F_5$ algorithms. We start with the basic theory of non-commutative polynomials, present the division algorithm, define Gröbner bases of ideals of non-commutative polynomials, and prove some of their fundamental properties. We continue with the classical Buchberger\u27s algorithm, introduce the concept of obstruction sets, and follow the steps of Teo Mora to the non-commutative version of the algorithm. Doing so, we use Dickson\u27s lemma to show that the original Buchberger\u27s algorithm terminates, and we prove the non-commutative version of Bucherger\u27s Criterion and correctness of the non-commutative version of Buchberger\u27s algorithm. We show how to transform sets of polynomials into matrices, and simultaneously formulate the commutative and non-commutative $F_4$ algorithm. We prove the correctness of the $F_4$ algorithm, and we prove it terminates if a finite Gröbner basis exists for the given ideal. For a finite set of polynomials, we define the syzygy module and prove some of its basic properties. We lift Buchberger\u27s theory into a free module over the ring of commutative polynomials and define polynomial signatures. We present the principal representative of the family of signature-based algorithms and prove its correctness and termination. We introduce the $F_5$ Criterion and use the signature-based algorithm to formulate the $F_5$ algorithm. We conclude by repeating the previous steps to arrive at a non-commutative signature-based algorithm and show the correctness of the non-commutative version of the $F_5$ algorithm. We prove this algorithm terminates if a finite Gröbner bases exists for the given ideal