44 строки
3.1 KiB
Markdown
44 строки
3.1 KiB
Markdown
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# Twin smooth integers
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This repository provides code to find *twin smooth integers*, which are two consecutive integers m and m+1 that both only have small prime factors.
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More precisely, an integer is *B-smooth* for some positive number B if all of its prime divisors are less than or equal to B. The number B is called the smoothness bound. *Twin B-smooth integers* are simply two consecutive numbers that are both B-smooth. A possible application of such pairs of numbers is that they can be used as parameters for some isogeny-based cryptosystems.
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## The PTE sieve
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The PTE sieve is a sieving algorithm to find twin smooth integers of a certain size with a desired smoothness bound. The algorithm uses a variant of the classical sieve of Eratosthenes that identifies smooth integers instead of primes. It is run on a search space of smaller integers. The output of the sieving step is then searched for specific patterns of smooth integers that correspond to solutions of the Prouhet-Tarry-Escott (PTE) problem. These solutions can be used to boost the pattern of smooth integers found among smaller numbers to twin smooth integers of the desired size.
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A detailed description of the method, the algorithm and results of a range of experiments are given in the paper
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[Sieving for twin smooth integers with solutions to the Prouhet-Tarry-Escott problem](https://eprint.iacr.org/2020/1283).
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The algorithm was implemented in Python 3 and has the option of running the main sieving procedure in C. See [pte_sieve/README](pte_sieve/README.md) for more details and instructions on how to use the code.
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### Contributors
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* Craig Costello
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* Patrick Longa
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* Michael Meyer
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* Michael Naehrig
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## Contributing
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This project welcomes contributions and suggestions. Most contributions require you to agree to a
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Contributor License Agreement (CLA) declaring that you have the right to, and actually do, grant us
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the rights to use your contribution. For details, visit https://cla.opensource.microsoft.com.
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When you submit a pull request, a CLA bot will automatically determine whether you need to provide
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a CLA and decorate the PR appropriately (e.g., status check, comment). Simply follow the instructions
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provided by the bot. You will only need to do this once across all repos using our CLA.
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This project has adopted the [Microsoft Open Source Code of Conduct](https://opensource.microsoft.com/codeofconduct/).
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For more information see the [Code of Conduct FAQ](https://opensource.microsoft.com/codeofconduct/faq/) or
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contact [opencode@microsoft.com](mailto:opencode@microsoft.com) with any additional questions or comments.
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## Trademarks
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This project may contain trademarks or logos for projects, products, or services. Authorized use of Microsoft
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trademarks or logos is subject to and must follow
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[Microsoft's Trademark & Brand Guidelines](https://www.microsoft.com/en-us/legal/intellectualproperty/trademarks/usage/general).
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Use of Microsoft trademarks or logos in modified versions of this project must not cause confusion or imply Microsoft sponsorship.
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Any use of third-party trademarks or logos are subject to those third-party's policies.
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