PreprintCC BY 4.0arXiv
Monogenic Fields of Cryptographic Size
Abstract
For a monic irreducible f ∈ ℤ[x] of degree n and an integer c, we study the palindromic transform F(x) = xn f(x + x−1 + c), which produces a polynomial of degree 2n. We give a discriminant formula disc(F) = f(c+2)f(c−2)disc(f)2, sufficient conditions for the irreducibility of F, and a criterion showing that F is monogenic when f is monogenic and f(c+2)f(c−2) is squarefree, together with a matching non-monogenicity criterion. Iterating the transform yields monogenic number fields of degree 2k n from a fixed base polynomial. As an explicit example, we construct a monogenic field of degree 512 from x4 + x + 1.
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BibTeX
@misc{adhikari2026monogenic,
title = {{Monogenic Fields of Cryptographic Size}},
author = {Swechchha Adhikari and Daphne Plott and Parker Torgersen},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.22374},
eprint = {2609.22374},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.22374},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Monogenic Fields of Cryptographic Size
AU - Swechchha Adhikari
AU - Daphne Plott
AU - Parker Torgersen
PY - 2026
DA - 2026/09/17
PB - arXiv
DO - 10.48550/arxiv.2609.22374
UR - https://arxiv.org/abs/2609.22374
AB - For a monic irreducible f ∈ ℤ[x] of degree n and an integer c, we study the palindromic transform F(x) = x^n f(x + x^−1 + c), which produces a polynomial of degree 2n. We give a discriminant formula disc(F) = f(c+2)f(c−2)disc(f)^2, sufficient conditions for the irreducibility of F, and a criterion showing that F is monogenic when f is monogenic and f(c+2)f(c−2) is squarefree, together with a matching non-monogenicity criterion. Iterating the transform yields monogenic number fields of degree 2^k n from a fixed base polynomial. As an explicit example, we construct a monogenic field of degree 512 from x^4 + x + 1.
ER -