[v3.0 — Ultrametric Synthesis] Complete analysis integrating three research threads: 1. Google Quantum AI and Oratomic's March 2026 Shor resource estimates for secp256k1 ECDLP 2. The ecdsa.fail crowdsourced optimization challenge (QWAV participation: scored 2,535,067,306, submitted to leaderboard) 3. QWAV's thermodynamic critique (Feynman-Shor Quantum Bifurcation, Post-Quantum Synthesis, SHORS IS BUSTED) NEW IN v3.0 (Section 10-11): Proposes ultrametric quantum computation as the alternative to Shor's algorithm. Introduces the Monna map (p-adic to real number readout), hierarchical error confinement via ultrametric topology, and prime factorization via geometric reconstruction instead of phase estimation. Proposes an ultrametric benchmark — p-adic modular exponentiation scored by branch operations × tree depth — as the next step for the field. Prior versions: v2.0 (DOI: 10.5281/zenodo.20517291), v1.1 (DOI: 10.5281/zenodo.20517264), v1.0 (DOI: 10.5281/zenodo.20517217).
[v3.0 — Ultrametric Synthesis] Complete analysis integrating three research threads: 1. Google Quantum AI and Oratomic's March 2026 Shor resource estimates for secp256k1 ECDLP 2. The ecdsa.fail crowdsourced optimization challenge (QWAV participation: scored 2,535,067,306, submitted to leaderboard) 3. QWAV's thermodynamic critique (Feynman-Shor Quantum Bifurcation, Post-Quantum Synthesis, SHORS IS BUSTED) NEW IN v3.0 (Section 10-11): Proposes ultrametric quantum computation as the alternative to Shor's algorithm. Introduces the Monna map (p-adic to real number readout), hierarchical error confinement via ultrametric topology, and prime factorization via geometric reconstruction instead of phase estimation. Proposes an ultrametric benchmark — p-adic modular exponentiation scored by branch operations × tree depth — as the next step for the field. Prior versions: v2.0 (DOI: 10.5281/zenodo.20517291), v1.1 (DOI: 10.5281/zenodo.20517264), v1.0 (DOI: 10.5281/zenodo.20517217).
[v2.0 — Full QWAV Thermodynamic Integration] Analysis of March 2026 Google Quantum AI and Oratomic papers on Shor's algorithm resource estimates for secp256k1 elliptic curve cryptography, presented alongside QWAV's published thermodynamic critique of Shor scalability. CONTENTS: - Resource estimates: Google (≤1,200 logical qubits, ≤90M Toffoli gates; <500K physical qubits superconducting) and Oratomic (10K physical qubits, neutral atoms) - ZK-proof censorship: First use in academic publishing; hidden optimizations independently rediscovered by Schrottenloher (arXiv:2606.02235) within two months - ecdsa.fail: Crowdsourced optimization challenge achieving 15.1% improvement over Google's classified circuit (135 submissions, 35 solvers) - Q-Day timeline: Justin Drake 50% by 2032 vs. QWAV "never" (thermodynamic impossibility) - QWAV THERMODYNAMIC CRITIQUE (new in v2.0): Full analysis of Landauer-limited QEC scaling, precision scaling problem, five false assumptions in Shor's algorithm, Feynman-Shor quantum bifurcation - Post-quantum migration: Ethereum Foundation's leanVM, 2029 target, Proximity Prize, Poseidon Initiative - UNRESOLVED DEBATE: Is Shor's algorithm physically viable at cryptographically relevant scales? Google/Oratomic: yes (engineering). QWAV: no (thermodynamic). No experimental resolution at scale exists. QWAV prior work integrated: Feynman-Shor Quantum Bifurcation (DOI: 10.5281/zenodo.17977869), Post-Quantum Synthesis (DOI: 10.5281/zenodo.17184229), SHORS IS BUSTED internal analysis.
Quantum Shor’s algorithm offers an exponential speedup over classical methods for integer factorization, a central problem to modern cryptography, and the implica- tions of Shor’s algorithm in factoring large numbers. The algorithm leverages on quantum parallelism to compute the period of a modular exponentiation function, employ- ing the Inverse Quantum Fourier Transform (IQFT) to extract periodicity, and classical post-processing to derive the factors. This final report of the Quantum Engineer- ing post-graduate course of the Fundacio Polit ́ enica de ` Catalunya, presents the main findings encountered in the implementation and benchmarking of Schor’s algorithm in real IBM quantum computers. To accomplish this, we created a command line tool made in Python, that allows the integration of the circuits implementing the Shor’s algorithm with the overall set of configuration parameters needed to perform the benchmark. In this report, we first provide an overview of the quantum factoring problem, followed by a detailed description of Shor’s algorithm, and its implementation using the Qiskit Python library. The circuit depth and noise effects are analyzed across different backend types, also different values for the optimization level and approximation degrees are configured when performing the circuit transpilation. As mentioned earlier different backend types were utilized for performance comparison, namely backend class with or without a noise model (AerSim), a fake backend class provided by the Qiskit library (fake provider), and the real quantum hardware (IBM Quantum Processing Units in January- June 2025).The study follows by analyzing the different error mitigation techniques: dynamic decoupling, Pauli twirling, the SABRE Optimization (Stochastic Algorithm for Quantum Boolean Optimization and Routing), and any combination of them. Once the optimum configuration is found, the maximum number that can be reasonably factorized with success is explored. Finally, the main conclusions, findings, and possible future research lines are presented, including the factorization of a N ≈ 1000 if appropriate parameters of the algorithm are selected.
Precise suites of benchmarks are required to assess the progress of early fault-tolerant quantum computers at economically impactful applications such as cryptanalysis. Appropriate challenges exist for factoring but those for elliptic curve cryptography are either too sparse or inadequate for standard applications of Shor's algorithm. We introduce a difficulty-graded suite of elliptic curve discrete logarithm (ECDLP) challenges that use Bitcoin's curve y^2=x^3+7 mod p while incrementally lowering the prime field from 256 down to 6 bits. For each bit-length, we provide the prime, the prime group order, and two deterministic nothing-up-my-sleeve (NUMS) points in compressed SEC1 form. All challenges are generated by a deterministic, reproducible procedure, and no private challenge scalar is chosen in advance. We calibrate classical cost against Pollard's rho records and quantum cost against resource estimation results for Shor's algorithm. We compile Shor's ECDLP circuit to logical counts and map them to physical resources for various parameters of the surface code, the repetition cat code and the LDPC cat codes. Under explicit and testable assumptions on physical error rates, code distances, and non-Clifford supply, our scenarios place the full 256-bit instance within a 2027--2033 window. The challenge ladder thus offers a transparent ruler to track fault-tolerant progress on a cryptanalytic target of immediate relevance, and it motivates proactive migration of digital assets to post-quantum signatures.
Today we witness an explosive growth in attention to Q-computing. Q-computing technologies, along with artificial intelligence (AI) and machine learning (ML) technologies, cloud and foggy computing, as well as technologies for collecting and streaming processing of Big Data and ETL, are constantly leading the lists of ""end-to-end"" information technologies for the digital economy of technologically developed countries of the world. One of the main reasons for this is the potential ability of quantum computers to solve some computational problems more efficiently than any of the most modern classical computers of the von Neumann architecture (supercomputers). The most expressive and interesting, from an applied point of view, examples of such problems are integer factorization, effectively performed by Shor's quantum algorithm, as well as record search in an unordered database, effectively solved by Grover's algorithm. This monograph contains the best practice for solving problems of quantum cryptanalysis to improve cyber security and resilience of the digital economy. The book discusses well-known and author's software implementations of promising quantum Shor algorithms, Grover, Simon et al. Shor's algorithm provides exponential acceleration of solving factorization problems, discrete logarithm problems (DLPs) and elliptic curve discrete logarithm problems (ECDLPs). The mentioned tasks are widely used in TLS, SSH or IPsec cryptographic applications of Internet/Intranet and IIoT/IoT networks, communication protocols based on Diffie–Hellman key agreements (dependent on the strength of the DLP or ECDLP), digital signature algorithms (DSA, ECDSA, RSA-PSS), public key encryption algorithms (El Gamal, RSA-OAEP), etc. In other words, Shor's quantum algorithm is potentially capable of violating these algorithms, and with them all the mechanisms of public-key cryptography deployed in cyberspace.
This analysis reconstructs the divergent phylogenetic lineages of quantum computing, termed Feynman-Shor bifurcation. The field has split into two distinct ontologies: path A (mimetic simulation), rooted in Feynman’s original 1982 vision of structural isomorphism, which is inherently robust to noise and viable on current hardware; and path B (algorithmic calculation), defined by Shor’s 1994 factoring algorithm, which remains practically intractable due to the fragility of phase estimation without fault tolerance. While this bifurcation defines the current era of hardware development, the two paths are projected to converge at the ultimate physical limits of scale and fidelity, resolving into a single, universal paradigm of quantum computation.