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Timothy Foo

Publications and source records attributed to Timothy Foo.

10 recordsLinked to original sources
arXiv2025

Brace for impact: ECDLP challenges for quantum cryptanalysis

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.

quant-ph↗
arXiv2016

On the sum of integers from some multiplicative sets and some powers of integers

We show that if there exists an integer subject to some congruence conditions that cannot be written as the sum of the norm of an ideal in $\mathbb{Z}[\exp(2\pi i/2^k)]$ and at most $k$ powers of $2$, $k\geq 3$, then there are infinitely many such integers. Also, if there exists an integer that cannot be written as the sum of an integer which is the norm of an ideal in in $\mathbb{Z}[\exp(2\pi i/p)]$ and at most $p-2$ powers of $p$, where $p\geq 3$ is a prime, then there are infinitely many such integers. Finally it is shown that there are infinitely many integers not the sum of the norm of an ideal in $\mathbb{Z}[\exp(2\pi i/p)]$ and at most $p-2$ powers of $p$, for $p\geq 3$ prime.

math.NT↗
arXiv2011

On the Discrepancy of the Roots of $x^2+1$ and $x^2+2$ to Prime Moduli

In this paper, we make a conjecture (conjecture 1) related to the Bateman-Horn conjecture and proceed to study the roots of $x^2+1$ and $x^2+2$ to prime moduli, assuming the truth of the Bateman-Horn conjecture and conjecture 1 and using the Erd\H{o}s-Turan-Koksma inequality.

math.NT↗