@phdthesis{20556, abstract = {Verifiable Delay Functions (VDFs) introduced by Boneh et al. (CRYPTO'18) are functions that require a prescribed number of sequential steps T to evaluate, yet their output can be verified in time much faster than T. Since their introduction, VDFs have gained a lot of attention due to their applications in blockchain protocols, randomness beacons, timestamping and deniability. This thesis explores the theory and applications of VDFs, focusing on enhancing their soundness, efficiency and practicality. The only practical VDFs known to date are based on repeated squaring in hidden order groups. Consider the function VDF(x,T)=x^(2^T). The iterated squaring assumption states that, for a random group element x, the result of VDF cannot be computed significantly faster than performing T sequential squarings if the group order is unknown. To make the result verifiable a prover can compute a proof of exponentiation (PoE) \pi. Given \pi, the output of VDF can be verified in time much less than T. We first present new constructions of statistically sound proofs of exponentiation, which are an important building block in the construction of SNARKs (Succinct Non-Interactive Argument of Knowledge). Statistical soundness means that the proofs remain secure against computationally unbounded adversaries, in particular, it remains secure even when the group order is known. We thereby address limitations in previous PoE protocols which either required (non-standard) hardness assumptions or a lot of parallel repetitions. Our construction significantly reduces the proof size of statistically sound PoEs that allow for a structured exponent, which leads to better efficiency of SNARKs and other applications. Secondly, we introduce improved batching techniques for PoEs, which allow multiple proofs to be aggregated and verified with minimal overhead. These protocols optimize communication and computation complexity in large-scale blockchain environments and enable scalable remote benchmarking of parallel computation resources. We then construct VDFs with enhanced properties such as zero-knowledge and watermarkability. It was shown by Arun, Bonneau and Clark (ASIACRYPT'22) that these features enable new cryptographic primitives called short-lived proofs and signatures. The validity of such proofs and signatures expires after a predefined amount of time T, i.e., they are deniable after time T. Our constructions improve upon the constructions by Arun, Bonneau and Clark in several dimensions (faster forging times, arguably weaker assumptions). Finally, we apply PoEs in the realm of primality testing, providing cryptographically sound proofs of non-primality for large Proth numbers. This work gives a surprising application of VDFs in the area of computational number theory. Together, our contributions advance both the theoretical foundations and the real-world usability of VDFs in general and in particular of PoEs, making them more adaptable and secure for current and emerging cryptographic applications.}, author = {Hoffmann, Charlotte}, issn = {2663-337X}, pages = {116}, publisher = {Institute of Science and Technology Austria}, title = {{Theory and applications of verifiable delay functions}}, doi = {10.15479/AT-ISTA-20556}, year = {2025}, } @article{10011, abstract = {We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478] - and hence any classical solution to mean curvature flow-is unique in the class of our new varifold solutions. This is in sharp contrast to the case of Brakke flows, which a priori may disappear at any given time and are therefore fatally non-unique. Finally, we propose an extension of the solution concept to the multi-phase case which is at least guaranteed to satisfy a weak-strong uniqueness principle.}, author = {Hensel, Sebastian and Laux, Tim}, issn = {1945-743X}, journal = {Journal of Differential Geometry}, keywords = {Mean curvature flow, gradient flows, varifolds, weak solutions, weak-strong uniqueness, calibrated geometry, gradient-flow calibrations}, pages = {209--268}, publisher = {International Press of Boston}, title = {{A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness}}, doi = {10.4310/jdg/1747065796}, volume = {130}, year = {2025}, } @inproceedings{20004, abstract = {A long-standing conjecture of Eckhoff, Linhart, and Welzl, which would generalize McMullen’s Upper Bound Theorem for polytopes and refine asymptotic bounds due to Clarkson, asserts that for k ⩽ ⌊(n-d-2)/2⌋, the complexity of the (⩽ k)-level in a simple arrangement of n hemispheres in S^d is maximized for arrangements that are polar duals of neighborly d-polytopes. We prove this conjecture in the case n = d+4. By Gale duality, this implies the following result about crossing numbers: In every spherical arc drawing of K_n in S² (given by a set V ⊂ S² of n unit vectors connected by spherical arcs), the number of crossings is at least 1/4 ⌊n/2⌋ ⌊(n-1)/2⌋ ⌊(n-2)/2⌋ ⌊(n-3)/2⌋. This lower bound is attained if every open linear halfspace contains at least ⌊(n-2)/2⌋ of the vectors in V. Moreover, we determine the space of all linear and affine relations that hold between the face numbers of levels in simple arrangements of n hemispheres in S^d. This completes a long line of research on such relations, answers a question posed by Andrzejak and Welzl in 2003, and generalizes the classical fact that the Dehn-Sommerville relations generate all linear relations between the face numbers of simple polytopes (which correspond to the 0-level). To prove these results, we introduce the notion of the g-matrix, which encodes the face numbers of levels in an arrangement and generalizes the classical g-vector of a polytope.}, author = {Streltsova, Elizaveta and Wagner, Uli}, booktitle = {41st International Symposium on Computational Geometry}, isbn = {9783959773706}, issn = {1868-8969}, location = {Kanazawa, Japan}, publisher = {Schloss Dagstuhl - Leibniz-Zentrum für Informatik}, title = {{Levels in arrangements: Linear relations, the g-matrix, and applications to crossing numbers}}, doi = {10.4230/LIPIcs.SoCG.2025.75}, volume = {332}, year = {2025}, } @article{19733, abstract = {One of the most striking quantum phenomena is superposition, where one particle simultaneously inhabits different states. Most methods to verify coherent superposition are indirect, in that they require the distinct states to be recombined. Here, we adapt an xor game, in which a “test” photon is placed in a superposition of two orthogonal spatial modes, and each mode is sent to separated parties who perform local measurements on their modes without reinterfering the original modes. We show that by using a second identical “measurement” photon the parties are nonetheless able to verify if the test photon was placed in coherent superposition of the two spatial modes. We then turn this game into a resource-efficient verification scheme, obtaining a confidence that the particle is superposed which approaches unity exponentially fast. We demonstrate our scheme using a single photon, obtaining a 99% confidence that the particle is superposed with only 37 copies. Our work shows the utility of xor games to verify quantum resources, allowing us to efficiently detect quantum superposition without reinterfering the superposed modes.}, author = {Kun, Daniel and Strömberg, Karl T and Spagnolo, Michele and Dakić, Borivoje and Rozema, Lee A. and Walther, Philip}, issn = {2469-9934}, journal = {Physical Review A}, number = {5}, publisher = {American Physical Society}, title = {{Direct and efficient detection of quantum superposition}}, doi = {10.1103/PhysRevA.111.L050402}, volume = {111}, year = {2025}, } @misc{22142, abstract = {Data to generate Figures from the Letter "Direct and Efficient Detection of Quantum Superposition" published in APS Physical Review A. The code can be found on the provided GitLab repository.}, author = {Kun, Daniel}, publisher = {Zenodo}, title = {{Direct And Efficient Detection of Quantum Superposition - Data and Figures}}, doi = {10.5281/zenodo.13375452}, year = {2025}, } @article{19636, abstract = {This summary of the second Terrestrial Very-Long-Baseline Atom Interferometry (TVLBAI) Workshop provides a comprehensive overview of our meeting held in London in April 2024 (Second Terrestrial Very-Long-Baseline Atom Interferometry Workshop, Imperial College, April 2024), building on the initial discussions during the inaugural workshop held at CERN in March 2023 (First Terrestrial Very-Long-Baseline Atom Interferometry Workshop, CERN, March 2023). Like the summary of the first workshop (Abend et al. in AVS Quantum Sci. 6:024701, 2024), this document records a critical milestone for the international atom interferometry community. It documents our concerted efforts to evaluate progress, address emerging challenges, and refine strategic directions for future large-scale atom interferometry projects. Our commitment to collaboration is manifested by the integration of diverse expertise and the coordination of international resources, all aimed at advancing the frontiers of atom interferometry physics and technology, as set out in a Memorandum of Understanding signed by over 50 institutions (Memorandum of Understanding for the Terrestrial Very Long Baseline Atom Interferometer Study).}, author = {Abdalla, Adam and Abe, Mahiro and Abend, Sven and Abidi, Mouine and Aidelsburger, Monika and Alibabaei, Ashkan and Allard, Baptiste and Antoniadis, John and Arduini, Gianluigi and Augst, Nadja and Balamatsias, Philippos and Balaž, Antun and Banks, Hannah and Barcklay, Rachel L. and Barone, Michele and Barsanti, Michele and Bason, Mark G. and Bassi, Angelo and Bayle, Jean Baptiste and Baynham, Charles F.A. and Beaufils, Quentin and Beldjoudi, Sélyan and Belić, Aleksandar and Bennetts, Shayne and Bernabeu, Jose and Bertoldi, Andrea and Bigard, Clara and Bigelow, N. P. and Bingham, Robert and Blas, Diego and Bobrick, Alexey and Boehringer, Samuel and Bogojević, Aleksandar and Bongs, Kai and Bortoletto, Daniela and Bouyer, Philippe and Brand, Christian and Buchmueller, Oliver and Buica, Gabriela and Calatroni, Sergio and Calmels, Léo and Canizares, Priscilla and Canuel, Benjamin and Caramete, Ana and Caramete, Laurentiu Ioan and Carlesso, Matteo and Carlton, John and Carman, Samuel P. and Carroll, Andrew and Casariego, Mateo and Chairetis, Minoas and Charmandaris, Vassilis and Chauhan, Upasna and Chen, Jiajun and Chiofalo, Maria Luisa Maria Luisa Marilù and Ciampini, Donatella and Cimbri, Alessia and Cladé, Pierre and Coleman, Jonathon and Constantin, Florin Lucian and Contaldi, Carlo R. and Corgier, Robin and Dash, Bineet and Davies, G. J. and De Rham, Claudia and De Roeck, Albert and Derr, Daniel and Dey, Soumyodeep and Di Pumpo, Fabio and Djordjevic, Goran S. and Döbrich, Babette and Dornan, Peter and Doser, Michael and Drougakis, Giannis and Dunningham, Jacob and Duspayev, Alisher and Easo, Sajan and Eby, Joshua and Efremov, Maxim and Elertas, Gedminas and Ellis, John and Entin, Nicholas and Fairhurst, Stephen and Fanì, Mattia and Fassi, Farida and Fayet, Pierre and Felea, Daniel and Feng, Jie and Flack, Robert and Foot, Chris and Freegarde, Tim and Fuchs, Elina and Gaaloul, Naceur and Gao, Dongfeng and Gardner, Susan and Garraway, Barry M. and Garrido Alzar, Carlos L. and Gauguet, Alexandre and Giese, Enno and Gill, Patrick and Giudice, Gian F. and Glasbrenner, Eric P. and Glick, Jonah and Graham, Peter W. and Granados, Eduardo and Griffin, Paul F. and Gué, Jordan and Guellati-Khelifa, Saïda and Gupta, Subhadeep and Gupta, Vishu and Hackermueller, Lucia and Haehnelt, Martin and Hakulinen, Timo and Hammerer, Klemens and Hanımeli, Ekim T. and Harte, Tiffany and Hartmann, Sabrina and Hawkins, Leonie and Hees, Aurelien and Herbst, Alexander and Hird, Thomas M. and Hobson, Richard and Hogan, Jason and Holst, Bodil and Holynski, Michael and Hosten, Onur and Hsu, Chung Chuan and Huang, Wayne Cheng Wei and Hughes, Kenneth M. and Hussain, Kamran and Hütsi, Gert and Iovino, Antonio and Isfan, Maria Catalina and Janson, Gregor and Jeglič, Peter and Jetzer, Philippe and Jiang, Yijun and Juzeliūnas, Gediminas and Kaenders, Wilhelm and Kalliokoski, Matti and Kehagias, Alex and Kilian, Eva and Klempt, Carsten and Knight, Peter and Koley, Soumen and Konrad, Bernd and Kovachy, Tim and Krutzik, Markus and Kumar, Mukesh and Kumar, Pradeep and Labiad, Hamza and Lan, Shau Yu and Landragin, Arnaud and Landsberg, Greg and Langlois, Mehdi and Lanigan, Bryony and Leone, Bruno and Le Poncin-Lafitte, Christophe and Lellouch, Samuel and Lewicki, Marek and Lien, Yu Hung and Lombriser, Lucas and Asamar, Elias Lopez and Lopez-Gonzalez, J. Luis and Lu, Chen and Luciano, Giuseppe Gaetano and Lundblad, Nathan and De J. López Monjaraz, Cristian and Lowe, Adam and Mackoit-Sinkevičienė, Mažena and Maggiore, Michele and Majumdar, Anirban and Makris, Konstantinos and Maleknejad, Azadeh and Marchant, Anna L. and Mariotti, Agnese and Markou, Christos and Matthews, Barnaby and Mazumdar, Anupam and Mccabe, Christopher and Meister, Matthias and Mentasti, Giorgio and Menu, Jonathan and Messineo, Giuseppe and Meyer-Hoppe, Bernd and Micalizio, Salvatore and Migliaccio, Federica and Millington, Peter and Milosevic, Milan and Mishra, Abhay and Mitchell, Jeremiah and Morley, Gavin W. and Mouelle, Noam and Müller, Jürgen and Newbold, David and Ni, Wei Tou and Niehof, Christian and Noller, Johannes and Odžak, Senad and Oi, Daniel K.L. and Oikonomou, Andreas and Omar, Yasser and Overstreet, Chris and Puthiya Veettil, Vishnupriya and Pahl, Julia and Paling, Sean and Pan, Zhongyin and Pappas, George and Pareek, Vinay and Pasatembou, Elizabeth and Paternostro, Mauro and Pathak, Vishal K. and Pelucchi, Emanuele and Pereira Dos Santos, Franck and Peters, Achim and Pichery, Annie and Pikovski, Igor and Pilaftsis, Apostolos and Pislan, Florentina Crenguta and Plunkett, Robert and Poggiani, Rosa and Prevedelli, Marco and Rafelski, Johann and Raidal, Juhan and Raidal, Martti and Rasel, Ernst Maria and Renaux-Petel, Sébastien and Richaud, Andrea and Rivero-Antunez, Pedro and Rodzinka, Tangui and Roura, Albert and Rudolph, Jan and Sabulsky, Dylan and Safronova, Marianna S. and Sakellariadou, Mairi and Salvi, Leonardo and Sameed, Muhammed and Sarkar, Sumit and Schach, Patrik and Schäffer, Stefan Alaric and Schelfhout, Jesse and Schilling, Manuel and Schkolnik, Vladimir and Schleich, Wolfgang P. and Schlippert, Dennis and Schneider, Ulrich and Schreck, Florian and Schwartzman, Ariel and Schwersenz, Nico and Sergijenko, Olga and Sfar, Haifa Rejeb and Shao, Lijing and Shipsey, Ian and Shu, Jing and Singh, Yeshpal and Sopuerta, Carlos F. and Sorba, Marianna and Sorrentino, Fiodor and Spallicci, Alessandro D.A.M. and Stefanescu, Petruta and Stergioulas, Nikolaos and Stoerk, Daniel and Thaivalappil Sunilkumar, Hrudya and Ströhle, Jannik and Tam, Zoie and Tandon, Dhruv and Tang, Yijun and Tell, Dorothee and Tempere, Jacques and Temples, Dylan J. and Thampy, Rohit P. and Tietje, Ingmari C. and Tino, Guglielmo M. and Tinsley, Jonathan N. and Tintareanu Mircea, Ovidiu and Tkalčec, Kimberly and Tolley, Andrew J. and Tornatore, Vincenza and Torres-Orjuela, Alejandro and Treutlein, Philipp and Trombettoni, Andrea and Ufrecht, Christian and Urrutia, Juan and Valenzuela, Tristan and Valerio, Linda R. and Van Der Grinten, Maurits and Vaskonen, Ville and Vázquez-Aceves, Verónica and Veermäe, Hardi and Vetrano, Flavio and Vitanov, Nikolay V. and Von Klitzing, Wolf and Wald, Sebastian and Walker, Thomas and Walser, Reinhold and Wang, Jin and Wang, Yan and Weidner, C. A. and Wenzlawski, André and Werner, Michael and Wörner, Lisa and Yahia, Mohamed E. and Yazgan, Efe and Zambrini Cruzeiro, Emmanuel and Zarei, M. and Zhan, Mingsheng and Zhang, Shengnan and Zhou, Lin and Zupanič, Erik}, issn = {2196-0763}, journal = {EPJ Quantum Technology}, publisher = {Springer Nature}, title = {{Terrestrial Very-Long-Baseline Atom Interferometry: Summary of the second workshop}}, doi = {10.1140/epjqt/s40507-025-00344-3}, volume = {12}, year = {2025}, } @article{12311, abstract = {In this note, we prove a formula for the cancellation exponent kv,n between division polynomials ψn and ϕn associated with a sequence {nP}n∈N of points on an elliptic curve E defined over a discrete valuation field K. The formula greatly generalizes the previously known special cases and treats also the case of non-standard Kodaira types for non-perfect residue fields.}, author = {Naskręcki, Bartosz and Verzobio, Matteo}, issn = {1473-7124}, journal = {Proceedings of the Royal Society of Edinburgh Section A: Mathematics}, keywords = {Elliptic curves, Néron models, division polynomials, height functions, discrete valuation rings}, number = {5}, pages = {1646--1660}, publisher = {Cambridge University Press}, title = {{Common valuations of division polynomials}}, doi = {10.1017/prm.2024.7}, volume = {155}, year = {2025}, } @article{20797, abstract = {Quantum key distribution (QKD) offers a theoretically secure method to share secret keys, yet practical implementations face challenges due to noise and loss over long-distance channels. Traditional QKD protocols require extensive noise compensation, hindering their industrial scalability and lowering the achievable key rates. Alternative protocols encode logical qubits in noise-resilient states but at the cost of using many physical qubits, increasing susceptibility to loss and limiting transmission distance. In this work, we introduce a logical-qubit encoding that uses antisymmetric Bell states in the continuous photonic degrees of freedom, frequency and time. By leveraging the continuous space, we overcome this noise-loss robustness trade-off by minimizing the number of photons per logical qubit while optimizing the encoding resilience over noise fluctuations. We analyze the security of our encoding and demonstrate its robustness compared to existing state-of-the-art protocols. This approach provides a path toward scalable, efficient QKD implementations under realistic noise conditions.}, author = {Seabrook, Hannah and Lavie, Emilien and Strömberg, Karl T and Stafford, Matthew P. and Rubino, Giulia}, issn = {2331-7019}, journal = {Physical Review Applied}, number = {2}, publisher = {American Physical Society}, title = {{Surpassing the loss-noise robustness trade-off in quantum key distribution}}, doi = {10.1103/xq2l-r4r7}, volume = {24}, year = {2025}, } @article{21244, abstract = {Given a family of varieties over the projective line, we study the density of fibres that are everywhere locally soluble in the case that components of higher multiplicity are allowed. We use log geometry to formulate a new sparsity criterion for the existence of everywhere locally soluble fibres and formulate new conjectures that generalise previous work of Loughran and Smeets. These conjectures involve geometric invariants of the associated multiplicity orbifolds on the base of the fibration in the spirit of Campana. We give evidence for the conjectures by providing an assortment of bounds using Chebotarev’s theorem and sieve methods, with most of the evidence involving upper bounds. }, author = {Browning, Timothy D and Lyczak, Julian and Smeets, Arne}, issn = {1944-7833}, journal = {Algebra & Number Theory}, number = {10}, pages = {2049--2090}, publisher = {Mathematical Sciences Publishers}, title = {{Paucity of rational points on fibrations with multiple fibres}}, doi = {10.2140/ant.2025.19.2049}, volume = {19}, year = {2025}, } @article{20850, abstract = {We provide an estimate for the number of nontrivial integer points on the Pellian surface t^2 - du^2 = 1 in a bounded region. We give a lower bound on the size of fundamental solutions for almost all d in a certain class, based on a recent conjecture of Browning and Wilsch about integer points on log K3 surfaces. We also obtain an upper bound on the average of class number in this class, assuming the same conjecture.}, author = {Diao, Yijie}, issn = {2118-8572}, journal = {Journal de theorie des nombres de Bordeaux}, number = {3}, pages = {973--988}, publisher = {Université de Bordeaux}, title = {{Class numbers and integer points on some Pellian surfaces}}, doi = {10.5802/jtnb.1348}, volume = {37}, year = {2025}, } @article{21003, abstract = {We extend work of Heath-Brown and Salberger, based on the determinant method, to provide a uniform upper bound for the number of integral points of bounded height on an affine surface, which are subject to a polynomial congruence condition. This is applied to get a new uniform bound for points on diagonal quadric surfaces, and to a problem about the representation of integers as a sum of four unlike powers.}, author = {Browning, Timothy D and Verzobio, Matteo}, issn = {2397-3129}, journal = {Discrete Analysis}, publisher = {Cambridge: Alliance of Diamond Open Access Journals}, title = {{Counting integer points on affine surfaces with a side condition}}, doi = {10.19086/da.143787}, volume = {2025}, year = {2025}, } @article{21260, abstract = {We prove that there does not exist F∈Q[x,y] of degree 4 such that F(Z^2 )=Z ≥0. In particular, this answers a question by John S. Lew and Bjorn Poonen for quartic polynomials.}, author = {Yao Xiao, Stanley and Yamagishi, Shuntaro}, issn = {1435-9863}, journal = {Journal of the European Mathematical Society}, publisher = {EMS Press}, title = {{Quartic polynomials in two variables do not represent all non-negative integers}}, doi = {10.4171/jems/1697}, year = {2025}, } @article{18822, abstract = {Let N(X) be the number of integral zeros (mathematical equation). Works of Hooley and Heath-Brown imply (mathematical equation), if one assumes automorphy and grand Riemann hypothesis for certain Hasse–Weil L-functions. Assuming instead a natural large sieve inequality, we recover the same bound on N(X). This is part of a more general statement, for diagonal cubic forms in (mathematical equation) variables, where we allow approximations to Hasse–Weil L-functions.}, author = {Wang, Victor}, issn = {2041-7942}, journal = {Mathematika}, number = {1}, publisher = {London Mathematical Society}, title = {{Diagonal cubic forms and the large sieve}}, doi = {10.1112/mtk.70008}, volume = {71}, year = {2025}, } @article{21768, abstract = {Let F∈Z[x1,…,xn] be a homogeneous form of degree d≥2, and V∗F the singular locus of the hypersurface {x∈AnC:F(x)=0}. A longstanding result of Birch states that there is a non-trivial integral solution to the equation F(x1,…,xn)=0 provided n>dimV∗F+(d−1)2d, and there is a non-singular solution in R and Qp for all primes p. We give a different formulation of this result. More precisely, we replace dimV∗F with a quantity HF defined in terms of the Hessian matrix of F. This quantity satisfies 0≤HF≤dimV∗F; therefore, we improve on the aforementioned result of Birch if HF3. Roughly speaking, to do so, we upgrade an order of magnitude result to a full asymptotic formula that was conjectured by Hooley in the number field setting.}, author = {Browning, Timothy D and Glas, Jakob and Wang, Victor}, issn = {1432-1823}, journal = {Mathematische Zeitschrift}, number = {4}, publisher = {Springer Nature}, title = {{Optimal sums of three cubes in Fq[t]}}, doi = {10.1007/s00209-025-03765-z}, volume = {310}, year = {2025}, }