Generating Random Factored Ideals in Number Fields

@article{Charles2018GeneratingRF,
  title={Generating Random Factored Ideals in Number Fields},
  author={Zachary B. Charles},
  journal={Math. Comput.},
  year={2018},
  volume={87},
  pages={2047-2056}
}
We present a randomized polynomial-time algorithm to generate a random integer according to the distribution of norms of ideals at most N in any given number field, along with the factorization of the integer. Using this algorithm, we can produce a random ideal in the ring of algebraic integers uniformly at random among ideals with norm up to N, in polynomial time. We also present a variant of this algorithm for generating ideals in function fields. 
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References

SHOWING 1-10 OF 14 REFERENCES

Generating random factored Gaussian integers, easily

We present a (random) polynomial-time algorithm to generate a random Gauss- ian integer with the uniform distribution among those with norm at most N, along with its prime factorization. The method

Generating Random Factored Numbers, Easily

  • A. Kalai
  • Computer Science, Mathematics
    SODA '02
  • 2002
TLDR
This work presents a significantly simpler algorithm and analysis for the problem of generating a random “pre-factored” number, that is, a uniformly random number between 1 and n along with its prime factorization.

How to Generate Factored Random Numbers

  • E. Bach
  • Mathematics, Computer Science
    SIAM J. Comput.
  • 1988
TLDR
This paper presents an efficient method for generating a random integer with known factorization, and can be implemented with randomized primality testing; in this case the distribution of correctly factored outputs is uniform, and the possibility of an incorrectly factored output can in practice be disregarded.

Efficient Factoring Polynomials over Local Fields and Its Applications

TLDR
The present results solve in particular problems posed by H.W. Lenstra, Jr. in [9].

PRIMES is in P

We present an unconditional deterministic polynomial-time algorithm that determines whether an input number is prime or composite.

Analytic methods in the analysis and design of number-theoretic algorithms

  • E. Bach
  • Computer Science, Mathematics
  • 1985
This book makes a substantial contribution to the understanding of a murky area of number theory that is important to computer science, an area relevant to the design and analysis of number-theoretic

Algebraic Number Theory

Notation Introduction 1. Algebraic foundations 2. Dedekind domains 3. Extensions 4. Classgroups and units 5. Fields of low degree 6. Cyclotomic fields 7. Diophantine equations 8. L-functions

Primality Testing with Gaussian Periods

TLDR
It turns out that the new polynomial time primality test due to Agrawal, Kayal, and Saxena can also be formulated in the Galois theory language, which leads to aPrimality test with a significantly improved guaranteed run time exponent.

Approximate formulas for some functions of prime numbers

Introduction to Algorithms

TLDR
The updated new edition of the classic Introduction to Algorithms is intended primarily for use in undergraduate or graduate courses in algorithms or data structures and presents a rich variety of algorithms and covers them in considerable depth while making their design and analysis accessible to all levels of readers.