Integer factoring and compositeness witnesses
We describe a reduction of the problem of factorization of integers n ≤ x in polynomial-time (log x)M+O(1) to computing Euler’s totient function, with exceptions of at most xO(1/M) composite integers that cannot be factored at all, and at most x exp −cM(loglogx)3(logloglogx)2$\begin{array}{} \disp...
| Main Authors: | , |
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| Format: | Article |
| Language: | English |
| Published: |
De Gruyter
2020-08-01
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| Series: | Journal of Mathematical Cryptology |
| Subjects: | |
| Online Access: | https://doi.org/10.1515/jmc-2019-0023 |
| Summary: | We describe a reduction of the problem of factorization of integers n ≤ x in polynomial-time (log x)M+O(1) to computing Euler’s totient function, with exceptions of at most xO(1/M) composite integers that cannot be factored at all, and at most x exp −cM(loglogx)3(logloglogx)2$\begin{array}{}
\displaystyle
\left(-\frac{c_M(\log\log x)^3}{(\log\log\log x)^2}\right)
\end{array}$ integers that cannot be factored completely. The problem of factoring square-free integers n is similarly reduced to that of computing a multiple D of ϕ(n), where D ≪ exp((log x)O(1)), with the exception of at most xO(1/M) integers that cannot be factored at all, in particular O(x1/M) integers of the form n = pq that cannot be factored. |
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| ISSN: | 1862-2976 1862-2984 |