Paper 2021/933
Fast Factoring Integers by SVP Algorithms, corrected
Claus Peter Schnorr
Abstract
To factor an integer $N$ we construct $n$ triples of $p_n$smooth integers $u,v,uvN$ for the $n$th prime $p_n$. Denote such triple a facrelation. We get facrelations from a nearly shortest vector of the lattice $\mathcal L(\mathbf R_{n, f})$ with basis matrix $\mathbf R_{n, f} \in \mathbb R^{(n+1)\times(n+1)}$ where $f\colon [1, n]\to[1, n]$ is a permutation of $[1, 2, \ldots, n]$ and $(f(1),\ldots,f(n),N'\ln N)$ is the diagonal and $(N' \ln p_1,\ldots, N' \ln p_n,N' \ln N)$ for $N' = N^{\frac{1}{n+1}}$ is the last line of $\mathbf R_{n, f}$. An independent permutation $f'$ yields an independent facrelation. We find sufficiently short lattice vectors by strong primaldual reduction of $\mathbf R_{n, f}$. We factor $N\approx 2^{400}$ by $n = 47$ and $N\approx 2^{800}$ by $n = 95$. Our accelerated strong primaldual reduction of [GN08] factors integers $N\approx 2^{400}$ and $N\approx 2^{800}$ by $4.2\cdot 10^9$ and $8.4\cdot 10^{10}$ arithmetic operations, much faster then the quadratic sieve and the number field sieve and using much smaller primes $p_n$. This destroys the RSA cryptosystem.
Metadata
 Available format(s)
 Category
 Publickey cryptography
 Publication info
 Preprint. Minor revision.
 Keywords
 Primaldual reductionSVPfacrelation
 Contact author(s)
 schnorr @ cs unifrankfurt de
 History
 20210709: received
 Short URL
 https://ia.cr/2021/933
 License

CC BY
BibTeX
@misc{cryptoeprint:2021/933, author = {Claus Peter Schnorr}, title = {Fast Factoring Integers by SVP Algorithms, corrected}, howpublished = {Cryptology ePrint Archive, Paper 2021/933}, year = {2021}, note = {\url{https://eprint.iacr.org/2021/933}}, url = {https://eprint.iacr.org/2021/933} }