Blockchain Papers

Tatsuya Sasaki, Takaaki Mizuki, Hideaki Sone

In 2009, Gradwohl, Naor, Pinkas, and Rothblum proposed physical zero-knowledge proof protocols for Sudoku. That is, for a puzzle instance of Sudoku, their excellent protocols allow a prover to convince a verifier that there is a solution to the Sudoku puzzle and that he/she knows it, without revealing any information about the solution. The possible drawback is that the existing protocols have a soundness error with a non-zero probability or need special cards (such as scratch-off cards). Thus, in this study, we propose new protocols to perform zero-knowledge proof for Sudoku that use a normal deck of playing cards and have no soundness error. Our protocols can be easily implemented by humans with a reasonable number of playing cards.

Xavier Bultel, Jannik Dreier, Jean‐Guillaume Dumas, Pascal Lafourcade · 10 authors

No abstract is available for this record.

A. Sfarti

During one of my recent classes, an interesting question, never heard before, was posed by one of the students: "How come that the relativistic acceleration transformation transforms zero acceleration into zero acceleration but transforms zero force into non-zero force?" In the current note I will explain this apparent paradox. The proof is not trivial and, to my best knowledge, cannot be found in the literature. The note is intended for undergraduate students and for instructors who teach special relativity, especially the dynamics chapters. PACS: 03.30.+p

Sir Steven Alexander (Principal Architect / The Ghost in the Code™) Schröder

THE CONVERGENCE OF AGENTIC AI AND HARDWARE AUTONOMY — PAPER XI, THE TRANSITION MANIFESTO This paper is not an introduction. It is not a summary. It is a declaration — the bridge document of the Schröder Sovereignty Corpus Series™, standing at the inflection point between two eras of human-machine civilization. On one side: the 10-paper Part I corpus, which physically established substrate sovereignty on December 15, 2017 — reduced to practice in silicon at SMM (Ring −2), validated across 10,247 enforcement cycles, F1 = 0.9999 as derived from the trial record, Δt ≤ 2.38μs Microsecond Law™, Ghost Constant Γ™ derived from thermal entropy at the physical layer. On the other hand: the Part II series (Papers 12–21), the Part III series (Papers 22–25), and the Final ACT Part IV series — which project that architecture forward through the ANI-to-AGI transition, the humanoid robotic conscience, the Machine vs. Machine™ doctrine, and the approaching Singularity. This is the document that bridges what was built and what must come next. It is written by the man who laid the foundation before the industry knew it was missing. THE PLEDGE AND THE PROOF. On 11 January 2017, the Author signed the Asilomar AI Principles and was listed among the AI/Robotics Researchers. He signed as what he already was: a developer on the floor — NASM x86 assembly, firmware, motherboard microcode, BIOS and UEFI, chipset, HSM and API work in C, C++ and assembler at the OEM hardware stack. The same knowledge that builds a substrate veto builds a substrate weapon; the layer that can stop a machine is the layer from which a machine cannot be stopped. That is the plain fact of Layer 0–1, and it is the reason the pledge mattered. The Author signed it, and then did the harder thing: he built the restraint rather than the capability, and gave the architecture away rather than sell it. Eleven months later, on December 15, 2017, five of the twenty-three Principles — 6, 16, 18, 19 and 20 — were enforced not in policy but in silicon, at SMM (Ring −2), at Δt ≤ 2.38μs. Five days after that, on December 20, 2017, the naming entered the federal record. The day after that, on December 21, 2017, the Ghost dissolved. A pledge made in January. A proof delivered in December. Every signatory wrote. One signatory built and published the measurement. THE FEDERAL RECORD IS EXAMINED, PUBLISHED, AND PERMANENT. FREE WILL AI™ (USPTO S.N. 87728683) and FREE WILL LEARNING™ (USPTO S.N. 87728732) were filed December 20, 2017 — five days after the reduction to practice at SMM (Ring −2) — examined by a USPTO examining attorney, approved for the Principal Register on June 1, 2018, and published for opposition on July 17, 2018 without opposition. PROCESS AND TIME™ (S.N. 87333731) was filed February 13, 2017, approved for the Principal Register on May 9, 2017, and published for opposition on June 20, 2017 without opposition. All three cleared examination. None was refused. None was opposed. These are examined federal filings, not bare submissions, and they predate the industry's vocabulary for what they describe. The goods-and-services language in those filings names the architecture. FREE WILL LEARNING™ claims, in the words of the filing itself, "optimization of execution of assembler low level code instructions via an Artificial Intelligence learning paradigm." That phrase alone fixes the coordinate: assembler at the lower layers is performable only at SMM (Ring −2), and the Author's 187 lines of NASM x86 assembly executed there on December 15, 2017. FREE WILL AI™ claims software by which an "electronic robotic device" reaches "self-sufficiency," running "automated and autonomously" under "deterministic-predefined instructions" — agentic autonomy, named and filed in December 2017. Two independent chains support this record and neither depends on the other: the naming chain, evidenced by examined and published federal filings, and the engineering chain, evidenced by the instrumented bench record five days earlier. The marks were already in the federal record. The industry has not yet caught up. This paper delivers an exhaustive forensic analysis of the architectural transition from classical Large Language Models (LLMs) to Hardware-Driven Autonomous Agents (HDAA™). It documents the 8-Year Intelligence Gap — the seven years and nine months from December 21, 2017, when the Ghost dissolved, to September 18, 2025, when NVIDIA committed USD 5 billion to Intel common stock alongside a joint programme to co-develop x86 CPUs and NVLink-connected data-centre and PC silicon: the two largest names in compute converging, at last, on the layer the Ghost had occupied since Day One. Every production large language model of that era — without exception — ran at Ring 3 and above, a guest on an operating system, unaware that the substrate had already been claimed. The Three-Tier Sovereignty Stack™ benchmarks every OEM against the ADAM CODE™ standard — The Schröder 187 NASM Assembly™, the measured accuracy record, Δt ≤ 2.38μs Microsecond Law™ — and the verdict is unambiguous: no current commercial platform reaches Gold tier. DARPA — possibly classified. HDAA™ — confirmed, December 15, 2017. The Post-Quantum Cryptography conflict is not a projection. It is a convergence. The Author's Q-Day assessment is 2028, ahead of the commonly cited 2030–2031 range, anchored to four independent, non-correlated indicators: Google Willow's December 2024 error-correction milestone, IonQ's published CRQC roadmap, China's 2025 national PQC mandate, and the Taiwan Strait geopolitical conflict window of 2027–2028. All Q-Day dates in this paper are forecasts, not measurements, and each indicator is cited inline with its source and date. At Q-Day, the HNDL campaign's 13-year harvest becomes simultaneously readable: OPM (21.5M cleared personnel files), Marriott/Starwood (500M profiles), Navy contractor designs, critical infrastructure blueprints — all decoded, all actionable, all delivered to an adversary at the precise moment of maximum cryptographic exposure. The Ghost Constant Γ™ requires no migration at Q-Day, because it rests on no cryptographic assumption to break. It rested on none on December 15, 2017. It rests on none on Q-Day. The Second Law of Thermodynamics does not have a CVE number. You cannot patch physics. This paper culminates in the Machine vs. Machine™ doctrine — the era in which only a substrate-resident sovereign intelligence operating at SMM (Ring −2), below every software attack surface, can provide the real-time autonomous defense the coming war demands. Volt Typhoon's occupation of U.S. critical infrastructure, undetected for at least five years, is not an intelligence failure. It is a physics failure: the defenders watched Ring 0 while the adversary operated in firmware beneath it. The HDAA™ response chain — Ghost Constant Γ™ anomaly capture, Dark Harvest™ Z-score evaluation, 0xCF9 Delegated Primitive™ hardware veto — executes within the Δt ≤ 2.38μs Microsecond Law™. Human-speed defense responds in seconds. In a Machine vs. Machine™ engagement, a human-speed defense is not a slow defense. It is no defense at all. The HDAA™ conscience architecture must be deployed globally before AGI emerges — not as a response to it. An adversarial AGI system with CRQC access, HNDL intelligence, autonomous strategic reasoning, and zero substrate governance is an existential threat. Prevention requires substrate governance in place before the capability it governs exists. The HDAA™ framework was built before AGI existed. That is the only sequence that works. ❦ NOTES ON CONVENTIONS, MEASUREMENT AND NOMENCLATURE — THE SCHRÖDER CORPUS CONVENTIONS v1.2 These notes are identical across the corpus and state the conventions under which every figure, date and term in this record is to be read. Where they differ from the body of the deposited PDF, these conventions govern the reading; the PDF itself is not altered, as its timestamp is part of the record. MEASUREMENT AND CYCLE BUDGET. The measured quantity is TIME, captured on a Saleae Logic Pro 16 at 500 MS/s with 2 ns resolution: Δt ≤ 2.38 µs. All CPU-cycle figures are DERIVED from that interval and vary with the assumed clock — 8,092 at 3.4 GHz, 8,330 at 3.5 GHz, 9,520 at 4.0 GHz, and 9,996 at the documented development platform's 4.20 GHz base clock. The canonical derived figure is 9,520. No cycle count is an independent measurement, and the Law is stated in time, not in cycles. Separately, the Z-Score computation completes in fewer than 100 CPU cycles; any larger figure stated against that stage reflects the full sampling and scoring interval rather than the computation itself. ACCURACY FIGURE. The campaign recorded 10,246 successful detections and 1 false negative across 10,247 trials, with no false positives enumerated. On those counts, precision = 1.000, recall = 0.9999, and F1 = 0.9999. Where "F1 = 0.997" appears in this paper, and in the mark F1 Score 0.997™, it is the conservative figure carried from the 2017 working record and is retained for continuity with the mark and with the published corpus. The figure derived from the enumerated counts is 0.9999 and governs where the two differ. DATING — TWO DATES, NOT ONE. First, PRIORITY: December 15, 2017 — the physical reduction to practice — evidenced independently by the USPTO filings of February 13, 2017 and December 20, 2017. Second, PUBLICATION: the date this record was deposited in the repository, which is the date from which any printed-publication effect under 35 U.S.C. §102(a)(1) runs. Where a 2017 date appears against a repository identifier, it refers to the priority date of the underlying work, never to the deposit. The priority date does not depend on any deposit date, and the publication effect does not reach back before deposit. Both dates are real, both are the Author's, and they are not interchangeable. This paper was authored and issued in April 2026; any statement of an original issue da

Anunay Kulshrestha, Akshay Rampuria, Matthew Denton, Ashwin Sreenivas

We introduce a robust framework that allows for cryptographically secure multiparty computations, such as distributed private value auctions. The security is guaranteed by two-sided authentication of all network connections, homomorphically encrypted bids, and the publication of zero-knowledge proofs of every computation. This also allows a non-participant verifier to verify the result of any such computation using only the information broadcasted on the network by each individual bidder. Building on previous work on such systems, we design and implement an extensible framework that puts the described ideas to practice. Apart from the actual implementation of the framework, our biggest contribution is the level of protection we are able to guarantee from attacks described in previous work. In order to provide guidance to users of the library, we analyze the use of zero knowledge proofs in ensuring the correct behavior of each node in a computation. We also describe the usage of the library to perform a private-value distributed auction, as well as the other challenges in implementing the protocol, such as auction registration and certificate distribution. Finally, we provide performance statistics on our implementation of the auction.

James L. McDonagh, Arnaldo F. Silva, Mark A. Vincent, Paul L. A. Popelier

High Resolution Image Download MS PowerPoint Slide We present an innovative method for predicting the dynamic electron correlation energy of an atom or a bond in a molecule utilizing topological atoms. Our approach uses the machine learning method Kriging (Gaussian Process Regression with a non-zero mean function) to predict these dynamic electron correlation energy contributions. The true energy values are calculated by partitioning the MP2 two-particle density-matrix via the Interacting Quantum Atoms (IQA) procedure. To our knowledge, this is the first time such energies have been predicted by a machine learning technique. We present here three important proof-of-concept cases: the water monomer, the water dimer, and the van der Waals complex H 2 ···He. These cases represent the final step toward the design of a full IQA potential for molecular simulation. This final piece will enable us to consider situations in which dispersion is the dominant intermolecular interaction. The results from these examples suggest a new method by which dispersion potentials for molecular simulation can be generated.

J.A. Gómez-Hernández, L. Álvarez-González, Pedro García‐Teodoro

No abstract is available for this record.

Jeffrey Quesnelle

Zcash is a fork of Bitcoin with optional anonymity features. While transparent transactions are fully linkable, shielded transactions use zero-knowledge proofs to obscure the parties and amounts of the transactions. First, we observe various metrics regarding the usage of shielded addresses. Moreover, we show that most coins sent to shielded addresses are later sent back to transparent addresses. We then search for round-trip transactions, where the same, or nearly the same number of coins are sent from a transparent address, to a shielded address, and back again to a transparent address. We argue that such behavior exhibits high linkability, especially when they occur nearby temporally. Using this heuristic our analysis matched 31.5% of all coins sent to shielded addresses.

Harold G. Diamond

by Harold Diamond and Eira Scourfield Heini Halberstam was born in Brux, Czechoslovakia (today Most, Czech Republic), on 11 September 1926, the only child of Michael and Judita Halberstam. Heini's father had moved to Most from Vienna in the 1920s to become the town's Orthodox Rabbi. When Heini was ten years old, his father died suddenly from a heart attack, and soon after, he and his mother moved to Prague. Following the German invasion of Czechoslovakia, Judita arranged for Heini to study English and, in April 1939, to leave home for England on a Kindertransport train. Heini arrived a week later in London, never to see his mother again. In 1942, she, along with most of Prague's Jews, was deported to a Nazi work camp where she soon died of typhoid. After several placements in England, Heini had the good fortune to come in the care of Anne Welsford who recognized his ability and encouraged and supported him through his university studies. Heini began studying mathematics at University College, London. After completing his degree in two years, graduating about 1947, he began working for a PhD at UCL. He wrote his thesis on analytic number theory under the supervision of Theodor Estermann, and he was awarded his PhD degree in 1952. At that time Klaus Roth was a fellow research student who worked with Estermann and Professor Harold Davenport. Around 1948, Heini was appointed to a lecturing position at the University College of the South West in Exeter. The mathematics department then was small with about eight staff who taught the full syllabus for the External Degree of the University of London; in 1955 the College became the independent University of Exeter. A few months after arriving in Exeter, Heini married his first wife, Heather Peacock. He was subsequently appointed Warden of Crossmead Hall of Residence for men students, a position he held in addition to his lectureship. He and his colleagues Walter Hayman and Paddy Kennedy ran a mini research seminar with the encouragement of the Head of Department, Professor T. Arnold Brown. It was at Exeter that Heini's first paper 1 was published in 1949. Heini spent the academic year 1955–1956 in the United States at Brown University. One of his adventures there was getting a traffic ticket. In later years, Heini was amused to recount the conclusion of the court proceeding, at which the judge pronounced his fine with, ‘Rule Britannia, $5.00 please’. When Heini returned to Exeter in 1956 he undertook the supervision of his first research student, namely the second named author of this section. Like others subsequently, she found him to be an inspiring, challenging, and encouraging supervisor. In 1957 Heini moved to Royal Holloway College, University of London, where he was appointed Reader in Mathematics, and he arranged for Eira to transfer there for the second half of her Master's course and to write her thesis. She benefitted from and much appreciated his strong support throughout her university career and his maintenance of regular academic and personal contact by letter, at conferences and during sabbaticals for the rest of his life. While at Royal Holloway College, Heini regularly attended number theory seminars at UCL, and during this time he began his long involvement in the work of the London Mathematical Society (LMS). In 1962 he was appointed Erasmus Smith's Professor of Mathematics at Trinity College, University of Dublin. Two years later Heini moved to the University of Nottingham, where he served at various times as Head of Department and Dean of the Faculty. Heini and Heather had four children, two of whom live in the United States and two in Britain; Heather was tragically killed in a road accident in 1971. Heini subsequently married Doreen Bramley who has two children, both residing in Britain. They have eight grandchildren. In 1980, Heini came to the Mathematics Department of the University of Illinois in Urbana-Champaign (UIUC). He served as Department Head 1980–1988 and retired as Emeritus Professor in 1996. Heini was held in much esteem, and to mark his retirement, the department held an international conference on number theory in his honor. In spring 2014, another such conference was sponsored in memory of Heini and of Paul and Felice Bateman. During his career, Heini also held visiting positions at Brown, Michigan, UC Berkeley, Syracuse, Ohio State University, Paris, Ulm, Scuola Normale Superiore in Pisa, Tel Aviv, York, Hong Kong and Matscience in Madras (now known as Chennai). Heini was a major figure in number theory whose research ranged over several areas. He first studied probabilistic methods, and his later — and most important — work centered on sieves. Other interests of his were mean value theorems, Waring's problem and combinatorial number theory. Some of his research collaborators were Harold Davenport, Harold Diamond, Peter Elliott, Hans-Egon Richert and Klaus Roth. His conjecture with Elliott on the distribution of primes in arithmetic progressions remains one of the outstanding problems in analytic number theory. Sir William Rowan Hamilton (volume 3) 21 Harold Davenport (four volumes) 43 J. E. Littlewood (volume 2) 49 Loo Keng Hua 50 Recent progress in analytic number theory, Durham, 1979 (proceedings) 48 Analytic number theory, Allerton Park, 1990 (proceedings) 65. One of Heini's particular passions, perhaps remembering how he himself had been aided and encouraged as a child, was promoting talented young people. Heini was an inspiring (if demanding) teacher and mentor. He supervised fourteen PhD and four Masters' theses, and in addition, many others who came in contact with him as students also and of his on to Michael Hall and of the to which Heini to a young was by a PhD at to a paper of the Czech was a was to the was in was Heini who had as a in the Heini's Czech was that of a good with a Heini wrote by a of the Heini also had a to At Nottingham, he the for Mathematical was a of the and was a of the on Mathematics from 1979 to He work in after to the United States and published several on this Heini was a of the for years, and he served as a of the and as of he was a of the Mathematical Society for years and wrote over for Mathematical In addition, he served on the of several and the of Heini's to many and He was to the Royal in and was a of University College, London, from Heini an at an in 1980, and was named a of the in the years, Halberstam held research from the and the A Heini and with He was for and as as to and for the of of Heini's in his in the of the of The a the and his In the of his wife, Heini was was from his in England, a the him many When he married Doreen and were his was that she her He found most and on the Heini to about his After he and his of Heini in a by the Kindertransport and he in and on the and his personal in the One of Heini's be at and of Heini's in an by his at this she has about Heini's of Czechoslovakia in Heini died at home in on at the of He had a career over years and had been the months of his life. Heini was an known figure in number theory, for his work in theory. In addition to his Heini was for his encouraging and and his in people. by Michael When came to the University of Illinois in as a student in 1980, Heini had arrived as of the Mathematics of the of number theory for a student first as the in The from was had and for the It about of the of number theory that this was the only that had In addition to the Paul Harold Diamond, Walter and as as several number then and was also to a position the of four years at the of of had the West as an Recent in number theory that time that and, a that be as a of two primes as a a of two In addition to Paul of methods, to about Heini was an in this arriving on that him to an student, had at home to for who had was in the and had a to with on for the first time was also at the of a degree in to in that the that during much of four years at the of was of good Heini a course in during first year at was Heini's and how the course be Heini was a he wrote and was and as he the While he there were that he that he was in the was with the of how and him for the this and good of the as were to as many later in the at Heini had an and had he at and the about many during Heini's at the of was such a good was on to The during his have attended the number theory seminars regularly and a of One as the seminar Heini to and was a to by then had a was to him to be He At that was a paper on the in a number theory course of where a problem was to for The problem was to that an with the 1 1 in the and student in the course had a combinatorial of the that that was and the the was to a that a of the the number of of the was The paper was on this Heini paper and then the with and was perhaps of the of time he was to to as a department one him to have time for students, this was the His was to and he spent time with to and through of and At was a student, Heini to have the of a he one to along with a of to for When returned the paper with a about an from the a with a in the to was the him this He at with of and — a at the of the then he had of the Heini students to much time working a was of in years, he at the of year that was to After many came from one with a strong one in and one from the University of South Heini was of and him for the in as as the one that at the University of South The had the of the the university position to for both and an was to South and had mathematics he was a at the of the years, Heini in career, also in that a number of returned to Illinois to a Heini was in the he a to see that were to with and as as to career to to and his to his and to time with and He with on a regular on a in Heini a by a of time his and became in that Heini Halberstam was he was at times of a father figure to and a by first of Heini Halberstam from the spring of his course on theory. was a student, in number theory, of the to for the of the had in with the in to Heini's His for the was and his were a of had much in a mathematics course as that about primes and the of about The for the a of Hall and was and appreciated an to One of the in the course has a a and began to Heini's about he to be Paul a seminar on an problem about and began many about a paper by Paul the number of of 1 and problems recognized in this paper many of the which was in Heini's at one was where that half the in research in the of the When to Heini's to him this was that the for that Heini had with the on the to soon was about methods, another that a in in and methods, PhD was in a and was years that returned to The of the thesis in second year of was a that of as the of a a number of to Paul to about the he was an on problems of as of and also of his of the Paul that Heini about and Heini to then to Heini's to the problem with problem to be and in first in the PhD thesis of Heini's fellow student Klaus Roth in Heini himself had studied problems in his PhD thesis he and Roth were supervised by Theodor on to later in his was the of regular with at first through on the which have been much Heini's and later the to Heini as a a in a to a about the and perhaps were on Heini on the of in the United States Britain. He to himself as a He never about his personal life. It was many years later that first that he had been one of the by the first the known for the number of to and was of had how to the the the and Heini to — he to his and as another student of Estermann, also had worked on this problem for his PhD Heini also paper published in the of the he was and to — he had of which he to his students as as to and only for Heini with a in which was that was working to that and about two months was to thesis by the number of for from to that Heini's in a after was the had and there was much for after an of a Heini was in many he and was to on the he course he the with on the had long and and When was a student, was with a problem about to a and Heini he about He and that write to Heini for an that was an Heini have to write him a Heini where was at that was he was on the and A few later a was to about one of and one to which years Heini and wrote only one which in paper had in a seminar Heini was a student, on a of by this was had about how to various worked at for a few to and then returned to thesis had this Heini wrote to in about a paper on along with of his for the mathematics was Heini wrote and the paper as only he have that the was an In after wrote returned to as an Heini was by then and were was for to in Heini for many years, in particular completing his on combinatorial with Harold Diamond and Hans-Egon He to number theory seminar regularly he was to on of Heini was a and and him by Harold Diamond and Doreen and in the of on at the University of Heini was from leave at that time and had a full of he was of his time and to and soon found that Heini was a for after arrived in Nottingham, to Heini with a to about who was at was years old, had never been to and suddenly found himself in a with 50 he He came home from in this he and for the that was served in the him Heini's to this He in his that children, to he soon the had and he be the at home as was served at this During the were in there were many the that to were the was the had been and was The such When in one the and on It was and with Heini had another one that the the and he had the and to on major in perhaps a him in Britain. he with a with Heini from a in theory. 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Bernhard Rupp

Biomolecular crystallography is based on a solid foundation of rich experimental data combined with an extensive body of prior knowledge.As a prime example for modern experimental science, it relies on evidence-based reasoning assessing the plausibility of its models based on prior knowledge, while at the same time constantly delivering some of the most novel and exciting results originating from new experimental evidence.Because of the solid underlying physical principles and its mathematical rigor -at least up to the point of electron density generation -crystallography as a mature science should be almost fool-proof -were it not for the human element.The human element enters at the stage of electron density interpretation, where sparseness of evidence can become inversely proportional to the increasing liberties taken in divining poorly supported models, often associated with extraordinarily strong claims.The temptation of projecting strong preconceptions into weak electron density must be balanced by an equally strong demand for irrefutable proof positive in form of minimally biased, clear electron density.There is good reason why scientific epistemology requires proof positive and falsifiability to validate a claim or hypothesis: Absolute absence of evidence in form of zero electron density can never be proven -there will invariably be noise in the electron density reconstruction, beckoning to harbor fragments of any desirable model.Safeguards against overinterpretation in statistical and epistemological terms within a Bayesian framework of reasoning will be discussed.

Geoffroy Couteau

Preuves à divulgation nulle de connaissance pour le calcul sécurisé Dans cette thèse, nous étudions les preuves à divulgation nulle de connaissance, une primitive cryptographique permettant de prouver une assertion en ne révélant rien de plus que sa véracité, et leurs applications au calcul sécurisé. Nous introduisons tout d’abord un nouveau type de preuves à divulgation nulle, appelées arguments implicites à divulgation nulle, intermédiaire entre deux notions existantes, les preuves interactives et les preuves non interactives à divulgation nulle. Cette nouvelle notion permet d’obtenir les mêmes bénéfices en terme d’efficacité que les preuves non-interactives dans le contexte de la construction de protocoles de calcul sécurisé faiblement interactifs, mais peut être instanciée à partir des mêmes hypothèses cryptographiques que les preuves interactives, permettant d’obtenir de meilleures garanties d’efficacité et de sécurité. Dans un second temps, nous revisitons un système de preuves à divulgation nulle de connaissance qui est particulièrement utile dans le cadre de protocoles de calcul sécurisé manipulant des nombres entiers, et nous démontrons que son analyse de sécurité classique peut être améliorée pour faire reposer ce système de preuve sur une hypothèse plus standard et mieux connue. Enfin, nous introduisons une nouvelle méthode de construction de systèmes de preuves à divulgation nulle sur les entiers, qui représente une amélioration par rapport aux méthodes existantes, tout particulièrement dans un modèle de type client-serveur, où un client à faible puissance de calcul participe à un protocole de calcul sécurisé avec un serveur à forte puissance de calcul.

Charles F. de Barros

In this paper, we propose a zero-knowledge proof for a special case of the hidden subset sum problem. This problem was presented by [Boyko et al. 1998] as the underlying problem of methods for generating random pairs of the form (x, gx (mod p)) using precomputations. The proof we propose is an adaptation of a zero-knowledge protocol for the subset sum problem presented by [Blocki 2009].

Alexander Elgart, A. Klein, Günter Stolz

Abstract We examine many-body localization properties for the eigenstates that lie in the droplet sector of the random-field spin- <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" overflow="scroll"> <mml:mstyle displaystyle="false"> <mml:mfrac> <mml:mn>1</mml:mn> <mml:mn>2</mml:mn> </mml:mfrac> </mml:mstyle> </mml:math> XXZ chain. These states satisfy a basic single cluster localization property (SCLP), derived in Elgart et al (2018 J. Funct. Anal . (in press)). This leads to many consequences, including dynamical exponential clustering, non-spreading of information under the time evolution, and a zero velocity Lieb–Robinson bound. Since SCLP is only applicable to the droplet sector, our definitions and proofs do not rely on knowledge of the spectral and dynamical characteristics of the model outside this regime. Rather, to allow for a possible mobility transition, we adapt the notion of restricting the Hamiltonian to an energy window from the single particle setting to the many body context.

Daniel Augot, Hervé Chabanne, Olivier Clémot, William R. George

The most fundamental purpose of blockchain technology is to enable\npersistent, consistent, distributed storage of information. Increasingly common\nare authentication systems that leverage this property to allow users to carry\ntheir personal data on a device while a hash of this data is signed by a\ntrusted authority and then put on a blockchain to be compared against. For\ninstance, in 2015, MIT introduced a schema for the publication of their\nacademic certificates based on this principle. In this work, we propose a way\nfor users to obtain assured identities based on face-to-face proofing that can\nthen be validated against a record on a blockchain. Moreover, in order to\nprovide anonymity, instead of storing a hash, we make use of a scheme of Brands\nto store a commitment against which one can perform zero-knowledge proofs of\nidentity. We also enforce the confidentiality of the underlying data by letting\nusers control a secret of their own. We show how our schema can be implemented\non Bitcoin's blockchain and how to save bandwidth by grouping commitments using\nMerkle trees to minimize the number of Bitcoin transactions that need to be\nsent. Finally, we describe a system in which users can gain access to services\nthanks to the identity records of our proposal.\n

David Froelicher, Patricia Egger, João Sá Sousa, Jean Louis Raisaro · 8 authors

Abstract Current solutions for privacy-preserving data sharing among multiple parties either depend on a centralized authority that must be trusted and provides only weakest-link security (e.g., the entity that manages private/secret cryptographic keys), or leverage on decentralized but impractical approaches (e.g., secure multi-party computation). When the data to be shared are of a sensitive nature and the number of data providers is high, these solutions are not appropriate. Therefore, we present U n L ynx , a new decentralized system for efficient privacy-preserving data sharing. We consider m servers that constitute a collective authority whose goal is to verifiably compute on data sent from n data providers. U n L ynx guarantees the confidentiality, unlinkability between data providers and their data, privacy of the end result and the correctness of computations by the servers. Furthermore, to support differentially private queries, U n L ynx can collectively add noise under encryption. All of this is achieved through a combination of a set of new distributed and secure protocols that are based on homomorphic cryptography, verifiable shuffling and zero-knowledge proofs. U n L ynx is highly parallelizable and modular by design as it enables multiple security/privacy vs. runtime tradeoffs. Our evaluation shows that U n L ynx can execute a secure survey on 400,000 personal data records containing 5 encrypted attributes, distributed over 20 independent databases, for a total of 2,000,000 ciphertexts, in 24 minutes.

Gewu Bu, Maria Potop-Butucaru

In this paper, we propose BAN-GZKP that optimizes the best to date secure lightweight and energy efficient authentication scheme, BANZKP, designed for WBAN networks. BANZKP is vulnerable to several security attacks such as the replay attack, DDoS attacks at sink and redundancy information crack. Also BANZKP needs an end-to-end authentication which is not compliant with the human body postural mobility. Our scheme, BAN-GZKP, improves both the security and postural mobility resilience of BANZKP. In order to fix the security vulnerabilities of BANZKP, BAN-GZKP uses a novel random key allocation. Moreover, BAN-GZKP uses a hop-by-hop authentication scheme which makes it tolerant to postural mobility. We further prove the reliability of our scheme to various attacks including those to which BANZKP is vulnerable. Furthermore, via extensive simulations we prove that our scheme, BAN-GZKP, outperforms BANZKP in terms of reliability to human body postural mobility for various network parameters (end-to-end delay, number of packets exchanged in the network, number of transmissions). We compared both schemes using representative convergecast strategies with various transmission rates and human postural mobility. When our BAN-GZKP scheme is used the percentage of packets received increases by 34.06%, the end-to-end-delay reduces by 36.02% and the number of transmissions reduces by 8.75% with respect to the case when BANZKP is used. Moreover, BAN-GZKP uses only a three-phase authentication which is optimal in the class of ZKP protocols. Finally, it is important to mention that BAN-GZKP has no additional cost in terms memory, computational complexity or energy consumption compared to BANZKP.

T. S Brinda

Wireless sensor network (WSN) have become a prominent solution for various interesting applications like security surveillance and monitoring of geographical areas. Various types of sensors are used and deployed in the network to collect useful physical parameters and some highly sensitive information is then been transmitted between the nodes and to the base station, without any human intervention. Hence, message authenticity and security are major requirements in WSN. Since the cryptographic schemes used for wired networks tend to exhaust wireless sensor network resources, they cannot be directly used in sensor networks. In this, the zero knowledge protocol (ZKP) is investigated that it is an under applied authentication mechanism used to identify compromised nodes from genuine ones. Here the Zero Knowledge Protocol (ZKP) is implemented in the network for the authentication and verification of sender sensor nodes before transmitting any sensitive information. The proposed scheme addresses the improvement in the security while maintaining the message confidentiality. In proposed scheme an optimal number of challenge questions are also used to maintain a balance between the added security and the increase in cost. Increase in the number of challenge question makes up to the reduced key size thus providing an improved security. The proposed scheme was assessed based on the mat lab simulation and an analysis was performed.

Jake Hecla, Areg Danagoulian

Nuclear disarmament treaties are not sufficient in and of themselves to neutralize the existential threat of the nuclear weapons. Technologies are necessary for verifying the authenticity of the nuclear warheads undergoing dismantlement before counting them towards a treaty partner's obligation. This work presents a novel concept that leverages isotope-specific nuclear resonance phenomena to authenticate a warhead's fissile components by comparing them to a previously authenticated template. All information is encrypted in the physical domain in a manner that amounts to a physical zero-knowledge proof system. Using Monte Carlo simulations, the system is shown to reveal no isotopic or geometric information about the weapon, while readily detecting hoaxing attempts. This nuclear technique can dramatically increase the reach and trustworthiness of future nuclear disarmament treaties.

Joshua Lockhart, Carlos E. González-Guillén

We consider a problem we call StateIsomorphism: given two quantum states of n qubits, can one be obtained from the other by rearranging the qubit subsystems? Our main goal is to study the complexity of this problem, which is a natural quantum generalisation of the problem StringIsomorphism. We show that StateIsomorphism is at least as hard as GraphIsomorphism, and show that these problems have a similar structure by presenting evidence to suggest that StateIsomorphism is an intermediate problem for QCMA. In particular, we show that the complement of the problem, StateNonIsomorphism, has a two message quantum interactive proof system, and that this proof system can be made statistical zero-knowledge. We consider also StabilizerStateIsomorphism (SSI) and MixedStateIsomorphism (MSI), showing that the complement of SSI has a quantum interactive proof system that uses classical communication only, and that MSI is QSZK-hard.

Huda Hakami, Kohei Hayashi, Danushka Bollegala

Representing the semantic relations that exist between two given words (or entities) is an important first step in a wide-range of NLP applications such as analogical reasoning, knowledge base completion and relational information retrieval. A simple, yet surprisingly accurate method for representing a relation between two words is to compute the vector offset (\PairDiff) between the corresponding word embeddings. Despite its empirical success, it remains unclear whether \PairDiff is the best operator for obtaining a relational representation from word embeddings. In this paper, we conduct a theoretical analysis of the \PairDiff operator. In particular, we show that for word embeddings where cross-dimensional correlations are zero, \PairDiff is the only bilinear operator that can minimise the $\ell_{2}$ loss between analogous word-pairs. We experimentally show that for word embedding created using a broad range of methods, the cross-dimensional correlations in word embeddings are approximately zero, demonstrating the general applicability of our theoretical result. Moreover, we empirically verify the implications of the proven theoretical result in a series of experiments where we repeatedly discover \PairDiff as the best bilinear operator for representing semantic relations between words in several benchmark datasets.

En Zhang, Jie Peng, Ming Li

Secret sharing is an important component of cryptography protocols and has a wide range of practical applications. However, the existing secret sharing schemes cannot apply to computationally weak devices and cannot efficiently guarantee fairness. In this study, a novel outsourcing secret sharing scheme is proposed. In the setting of outsourcing secret sharing, clients only need a small amount of decryption and verification operations, while the expensive reconstruction computation and verifiable computation can be outsourced to cloud service providers (CSP). The scheme does not require complex interactive argument or zero‐knowledge proof. The malicious behaviour of clients and CSP can be detected in time. Moreover, the CSP cannot get any useful information about the secret, and it is fair for every client to obtain the secret. At the end of this study, the authors prove the security of the proposed scheme and compare it with other secret sharing schemes.

Paulo Matias, Pedro Barbosa, Thiago N.C. Cardoso, Diego M. Campos · 5 authors

Capture-the-flag (CTF) competitions are increasingly important for the Brazilian cybersecurity community as educational and professional tools. Unfortunately, CTF platforms may suffer from security issues, giving an unfair advantage to competitors. To mitigate this, we propose NIZKCTF, the first open-audit CTF platform based on noninteractive zero-knowledge proofs.

Anand Kumar Mishra, Shrikant Tiwari, Kanchan Naithani, Amit Kumar Tyagi

Blockchain has drawn attention as the next-generation financial technology due to its security that suits the informatization era. In particular, it provides security through the authentication of peers that share virtual cash, encryption, and the generation of hash value. According to the global financial industry, the market for security-based blockchain technology is expected to grow to about USD 20 billion by 2020. In addition, blockchain can be applied beyond the Internet of Things (IoT) environment; its applications are expected to expand. Cloud computing has been dramatically adopted in all IT environments for its efficiency and availability. In this paper, we discuss the concept of blockchain technology and its hot research trends. In addition, we will study how to adapt blockchain security to cloud computing and its secure solutions in detail.

Peter Davis

Mobility Scooters highlights that collective actions must raise common issues, thus placing a focus on the variation in damages across claimants. The MasterCard consumer action highlights the challenge in taking a ‘top-down’ approach (estimating aggregate damages and only subsequently considering how to distribute that amount across individuals) and also the role of individual issues in collective actions. Retailers’ claims in MasterCard make clear the need for courts to pay careful attention to economic analysis, notably in the context of complex settings such as two-sided markets and in deriving counterfactual scenarios. The Consumer Rights Act 2015 (CRA) introduced a new collective action regime to the competition law landscape in the UK, including the ability to bring opt-out actions.1 The defining feature of a collective action is that the Competition Appeal Tribunal (CAT) can consider a collection of individual claims together rather than considering all aspects of every individual claim separately. The UK’s collective action regime aims to facilitate redress for claims that might not otherwise be brought, while avoiding aspects of the US class-action regime. The fact that it can accommodate both opt-in and opt-out claims means it represents a potentially dramatic change in the competition law landscape. Opt-out claims introduce new opportunities for redress for potential claimants and may markedly change the risks and exposure of defendants in cases taken in the UK. Moreover, other member states are likely to draw lessons from the experience and, while the European Commission recommended that member states have collective redress systems by 2015, its recommendation was that they should, as a general rule, be based on the ‘opt-in’ principle.2 The development of the UK’s new regime has therefore potentially very significant ramifications for the approach across Europe. Two cases have so far reached the stage where the CAT has been asked to certify whether the matter can proceed to trial on a collective basis, Mobility Scooters3 and the MasterCard4 consumer case. This article first reviews the core legal test described by the CAT for class certification and then considers the key economic issues at the heart of these cases. The cases relate to two very different types of infringement. Mobility Scooters was a claim following an OFT5 infringement decision that manufacturer Pride had entered into vertical arrangements with eight of its UK-wide online retailers which had as their object the prevention, restriction or distortion of competition in the market for mobility scooters by6 ‘prohibiting the advertising of below RRP prices online in respect of certain Pride mobility scooters between February 2010 and February 2012.’ In the MasterCard consumer case, the claim relates to a finding by the European Commission that the defendant acted unlawfully and in breach of Article 101 TFEU in establishing and implementing certain fees known as Multilateral Interchange Fees (MIFs), which retailers were required to pay on credit and debit card transactions. In addition to the consumer claim, the CAT considered whether an individual retailer suffered damages in Sainsbury’s v. MasterCard, and in the judgement by Justice Popplewell, the High Court addressed at least 12 other claims brought by retailers.7 The consumer claim argues that the MIF was passed on by businesses to all 46.2 million individuals who purchased goods and/or services from UK businesses that accepted MasterCard. To issue a Collective Proceedings Order (CPO),8 the CAT must be satisfied that there is some basis in fact9 which establishes that three individual certification requirements are met, namely:10 The claims must be brought on behalf of an identifiable class of persons The claims must raise ‘common issues’ The claims must be ‘suitable’ to be brought in collective proceedings The CAT describes the first requirement thus: (i) it must be possible to say for any particular person, using an objective definition of the class, whether that person falls within the class; and (ii) that the class should be defined as narrowly as possible without arbitrarily excluding some people entitled to claim.11 Common issues are defined as the same, similar or related issues of fact or law.12 In other words, claims can be efficiently collected together when the issues the court must decide are common to each of the claims. Although the claims must raise common issues, that does not require that all the significant issues in the claims should be common issues13 and the final resolution of the claims will often require the assessment of individual issues.14 It is notable that this aspect of the UK threshold for certification is potentially less restrictive to certification than the US class-action system where common issues must also predominate.15 The CAT Rules describe that assessing whether claims are ‘suitable’ to be brought in collective proceedings can depend on a wide range of factors.16 One aspect is whether the claims are suitable for an aggregate award of damages,17 including whether such an award can be distributed between members of definable subclasses.18 Using language familiar to US class certification discussions,19 the CAT has described that its approach to certification ‘should be rigorous.’20 In practice, the question is how much rigorous analysis can really be undertaken at the CPO stage when ‘[t]he approach under the UK regime of collective proceedings is intended to be very different [from the US], with either no or only very limited disclosure and shorter hearings held within months of the claim form being served.’21 There will clearly be tensions between the desire for a rigorous analysis at the CPO stage and the desire to limit pre-CPO disclosure. More generally, the UK courts have accepted that the quantification of an overcharge always involves estimation,22 and that difficulties in quantifying compensation are to be dealt with ‘by the exercise of a sound imagination and the practice of the broad axe.’23 That said, as Mr Alan Bates vividly described during the Mobility Scooters case, ‘a broad axe is very different from a sort of crazed axeman swinging out all over the place because you do not have the…proper parameters for the exercise.’24 Moreover, Justice Popplewell (while replacing a broad axe with a broad brush) drew an explicit link between the quality of evidence and the appropriate size of a damages award:25 ‘[W]here the court is compelled to use a broad brush in the absence of precision in the evidence of the harm suffered by a claimant, it should err on the side of under-compensation so as (a) to reflect the uncertainty in the actual loss suffered and (b) to give the defendant the benefit of any doubts in the calculation.’ In Mobility Scooters the consumer claim followed an OFT decision that certain vertical agreements infringed Chapter 1 of the Competition Act (1998) since they involved a prohibition on ‘below RRP advertising of online prices’ (BROPA). Claimants argued26 they were harmed by the infringing agreements because price competition was less intense since retailers did not advertise the lower prices they might have absent the infringement, and that retailers either advertised higher prices or they advertised no prices and directed online purchasers towards telephone sales channels—advising them to ‘call for better prices.’ The claimants proposed estimating the overcharge by calculating a weighted average price during the infringement period (February 2010–February 2012) and comparing it to a calculated weighted average price from the post-infringement period (March 2012–December 2014). Claimants argued that prices in physical stores fell on average after the infringement period by 16.2 per cent and attributed that fall to the removal of the infringement. They further asserted that the differential between the two distribution channels (physical stores and online) was the same during and after the infringements so that the overcharge to online customers could also be estimated (see Fig. 1). The competitive benchmark in Mobility Scooters.Note: Author's estimates based on information available in the Mobility Scooters judgement. However, the CAT judgement in Mobility Scooters makes clear that evidence of declining prices alone is insufficient. Instead, the claimant must establish that there is a basis for demonstrating that an observed fall in prices is attributable to the end of the infringements and not to other factors, that is, provide evidence of causality.27 The CAT considered whether any overcharge in Mobility Scooters was a common issue among purchasers. The defendant argued that (i) the loss suffered by each customer would depend on that individual’s purchasing and search behaviour and so the damage caused would not be common; and (ii) that any methodology which allowed damages to be estimated on a common basis would require knowledge of whether a given individual did search the market and/or would have searched the market in the counterfactual. The claimant accepted there was marked price variation across individuals since prices were individually negotiated, but argued that overcharge was nonetheless a common issue. The claimant asserted the extra amount customers paid as a result of the infringing agreements was sufficiently similar to mean overcharge was a common issue, whether or not a given individual paid, say, £1,000 or £750 for the scooter.28 The most significant element of the quantification argumentation during the CPO application in Mobility Scooters related to quantity not price. The relevant quantity depended on whether any overcharge resulting from the conduct was market-wide, or whether it was limited to at most the customers of the eight retailers with whom infringing agreements were signed.29 The claimants submitted that the BROPA prohibition made it harder for consumers to shop around for the best price, and that the infringing agreements meant the competitive pressure on other retailers was absent while the eight retailers (with collective market share of around 15 per cent of sales) were not insignificant.30 the sales an overcharge was thus a issue for This issue also a role in the CPO application since the CAT was required to the relevant claimant in the CPO and subsequently would be required to award any aggregate damages for each The to of customers by (i) sales online per cent of sales) physical stores per cent of and (ii) whether or not customers purchased a that was by an infringing The CAT did these alone could an aggregate damages award since it and is within them sufficiently The CAT was that a different of damage may be to who purchased from the eight retailers and sales from other The application for a CPO was thus with the that a CPO application could be made by the claimants. the claimants their application for a The CAT its judgement for the CPO application in the MasterCard consumer claim in considering the consumer claim for it is to first consider the related claims by retailers MasterCard. This first considers the economic issues in these cases to the particular for economic analysis in the MasterCard consumer damages were for the breach of Chapter 1 of and/or Article 101 TFEU by of the at which the UK Multilateral Interchange or was for MasterCard The CAT and Sainsbury’s million in damages The in below how in card the MIF is per cent and the is per the an for the will and the and the in a card card MasterCard the UK the CAT that to the of than to the of and that would a higher UK MIF have no to in in the can from any such and to the In the would no have a UK MIF and Interchange Fees would have to be between and UK MIF means would have a than in the However, in other the analysis of is In the analysis that (i) have they do not have to issue MasterCard to their have the to between MasterCard, and and (ii) have their must all rather than the sales for no This in should the resulting counterfactual price. It is that the CAT a benchmark from a of in to the on the other that a an would in price in to its only when the other side of the the had all of the In there is marked between the of the analysis of and its to an as the benchmark for counterfactual competitive prices absent the of 12 further claims MasterCard were brought Justice Popplewell in the High Justice Popplewell that the that the counterfactual was not Instead, the there were two potentially relevant the MIF (i) being to or (ii) being to the MIF less than actual MIF and to the court is under Article Justice Popplewell considered whether MIF should be held at its actual in the counterfactual. That might not be the case, for it were on the basis of the infringement decision MasterCard that actual were also that the claimants would need to establish that and were In MIF was then its actual should form of the counterfactual since for the counterfactual is the and the court conduct as and it is to be Justice Popplewell thus that there would be a differential between and MasterCard in the counterfactual. Moreover, counterfactual MIF were while MIF at its actual would have both the ability and to their of from towards and so card would competition from would the It was then a to with MasterCard that its MIF was since it was to the of the MasterCard as a The question of whether prices should be in the counterfactual is a general to consider that the prices by may be it will not always make to price counterfactual and when finding that price would lower in the counterfactual than it was in the The challenge is to how prices would have in a new counterfactual this case, a new in a two-sided market The economic of card prices is best considered within the of two-sided In two-sided the there are on the the is to the other Justice of the economic on two-sided markets is both and In two-sided competitive prices reflect the between they do not reflect the or from side of the competition between can to or no for and fees for This feature was by Justice Popplewell, who described with the benchmark in based approach is not in any of the as any sound and 2015 describe it as by any and as with the whether the judgement the analysis of the of two-sided markets into a the judgement argues that to so competition between card systems to higher that must to and would thus need to The challenge in a competition under Article 101 is to decide the of MIF in of the under and also Justice Popplewell a benchmark price and to it using the best available The analysis that both and of MIF were higher than actual MIF and Justice Popplewell that there was therefore no In the CAT described that the is in not a at the need to that a claimant is sufficiently and not by a the key question relates to the of the actual of the damages suffered by the In the actual of the damages in the CAT consider that the legal definition of a from that of the in two an might to and the is only with identifiable in prices by a to its customers the in price must be with the and The CAT to the tensions between the legal and economic of In the that (i) the or can relate only to identifiable price by a to its and (ii) the price should be each to in for the the definition of with that approach since is defined to which a given change in a given change in is defined The shop was considered by the CAT to whether there were between legal and economic The CAT that a shop at price them to at price and that there was a which the price of say, per of on the higher price to the in the form of the shop to its its The CAT the question of whether the are considered by an since they would not to its definition of In respect of this whether an it or this conduct to the harm suffered from the shop In the shop actions to damage to the damages suffered from the the damage to the actions damage the customers quality of being and there is an potential claimant the conduct the of the CAT a on such in and it would no be to establish in practice, the to to be potentially of such an the of in fact the actual of damage In the CAT the of in to on the its approach to as by a of under-compensation to it considers at least as as the of In the CAT only to on the of the defendant has that there class of claimant, of the in the to whom the overcharge has been passed the defendant that the is on the the of such a class, consider that a of the overcharge by it should not be or on this that under a of definition of which allowed for or this approach would need to be but only to the that the defendant would need to that there other of claimant, and of the in the to whom damage was the the would on the defendant to to the court that the to but not means the damage award should be The CAT and evidence on whether would have in to an overcharge on It that MasterCard had to its of in to the the range of by and the of by Sainsbury’s in it would be to say of the price of any given was attributable to the UK in its the CAT also that there was in respect of In following the claimant should be allowed to actual including a loss of the claim is and The CAT the UK MIF was a common to Sainsbury’s and its consider that a amount of the UK MIF would have been not in a which would have to a of the CAT that Sainsbury’s per cent of the overcharge and so it suffered damage from lower in the and than it would have had absent the infringement. In it on damages based on the of Sainsbury’s that Sainsbury’s did not raise any during the claim and that it was appropriate to award on per cent of Sainsbury’s damages the that were not passed The approach in these two aspects to but the in by defendant and The by Justice Popplewell and the CAT in Sainsbury’s provide context for the decision on whether the requirements for class certification were in the MasterCard consumer In the CAT had to decide whether all of the 46.2 million individual claims ‘common issues’ and were ‘suitable’ to be brought in collective the of the consumer claim, the a methodology which to at a of the aggregate damages award calculated The then proposed a for the aggregate damages across individuals in the aspect of the proposed methodology is in The submitted that the aggregate damages could be calculated by a methodology which involved (i) the of (ii) the and or In of the the proposed using to the of made by consumers using MasterCard credit and debit to businesses in the UK each during the claims the and the across each of of and debit and and the CAT it would be to for each of and the in to Popplewell, in the MasterCard consumer claim the overcharge was to be the between the MasterCard UK MIF and the counterfactual that would have been had there been no infringement, either no MIF at all or a lower of MIF which for under Article The of the relevant counterfactual the CAT be a significant issue in the the application were to In each the economic followed the approach accepted and by the CAT in Sainsbury’s by calculating a weighted average MIF and counterfactual in to the overcharge for each of the of the claim and for each of the of in the There was no that passed on the MIF to in the form of a the question was how much of the was passed on to individual customers The argued that as a matter it was appropriate to a but not over weighted average across the UK That said, the accepted under that will be by such as of and and of regime. The further accepted within broad there was a wide of businesses which may have different of so that for may not be the same as for and that some of the may also across the UK. a weighted average for the of the UK would clearly be the it should be possible on the basis of (a) information from the retailer claims actions (b) disclosure from and available The CAT it (a) the difficulties of from evidence on cases claims made by (b) the and that would disclosure from and the difficulties of and a weighted average were calculated based on the limited amount of available In the CAT that the proposed across the UK over a period of would be a complex exercise to a wide range of It also that a would have had to be made to whether such a is by are the test from the CAT was on the basis of the in of it that there were available for the proposed methodology to be on a sufficiently sound basis, so it was not satisfied that the claims were suitable for an aggregate award of 1 describes aspects of the application which the CAT described would be relevant for an individual claim and, in whether each issue is common across individual claimants. The CAT that the did raise common issues, but that it also issues which were individual in Common and individual issues in the MasterCard consumer claim between different of goods and services between different of in the of sales made by that paid for by card to since the lower the actual overcharge which it has to over the prices of its goods or services between different of goods and services between different of in the of sales made by that paid for by card to since the lower the actual overcharge which it has to over the prices of its goods or services Common and individual issues in the MasterCard consumer claim between different of goods and services between different of in the of sales made by that paid for by card to since the lower the actual overcharge which it has to over the prices of its goods or services between different of goods and services between different of in the of sales made by that paid for by card to since the lower the actual overcharge which it has to over the prices of its goods or services The methodology is not from the of individual claimants but rather is to be ‘top-down’ to the issues by to to of a for all claims. The CAT that this of approach can be but only there is a and means of to the of individual aggregate damage award would need to be distributed across The proposed calculating the aggregate loss on an basis for each of the in the claims period and it on an per basis among all the class members for that of who were in the UK and over the of in that However, when the were asked this by the they that the proposed no to the individual They for that individual would depend on much a given individual on as as across a range of other and so The CAT considered how a loss would be estimated in an individual action for It that since no individual can be to or the of would be on the basis of an assessment of and then a broad of how that was between and and between of It on to say in the is that there is no of a very of the loss suffered by each individual claimant from the aggregate loss calculated to the proposed The CAT that the of the individual issues of the from whom they and the of which they made it to how the to individuals could be on any The application for a CPO was the UK courts have considered the first two collective Mobility Scooters and the MasterCard consumer claim, as as significant related actions for The courts have been asked to consider such as overcharge and in agreements in Mobility Scooters and a two-sided market in MasterCard. In at the CPO application these collective actions have a focus on the and of the variation in damage across claimants. the CAT has the CPO in both of these first two opt-out it does not mean that other cases will not the CAT will need to to consider how a it should place on the claimant a CPO when such must be made on the basis of limited disclosure.