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Elliptische Kurven in der Post-Quantum-Kryptographie

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Zusammenfassung

In diesem Artikel geben wir eine kurze Einführung zu aktuellen Themen der Kryptographie, insbesondere zur Post-Quantum-Kryptographie. Ausgehend von gebräuchlichen Verfahren mit elliptischen Kurven erklären wir dabei, wie Isogenien zwischen elliptischen Kurven als Basis für neue Verfahren eingesetzt werden können. Dies findet im anschließend vorgestellten SIDH-Verfahren Anwendung, welches auch im aktuellen Standardisierungsprozess des National Institute of Standards and Technology (NIST) in leicht abgewandelter Form zur Debatte steht.

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Notes

  1. Letzteres wird als Diffie-Hellman-Problem bezeichnet.

  2. Dies wird beispielsweise in [6] ausgenutzt, um die Schlüsselgröße in SIDH zu verringern.

  3. Weitere Informationen, u. a. zu allen eingereichten Verfahren, finden sich unter https://csrc.nist.gov/projects/post-quantum-cryptography/.

  4. D. h. dass jede supersinguläre elliptische Kurve über \(\mathbb{F}_{p^{k}}\) isomorph zu einer supersingulären Kurve über \(\mathbb{F}_{p^{2}}\) ist.

  5. Anschaulich gesprochen haben diese die Eigenschaft, dass ausgehend von einem Knoten eine „große“ Menge von anderen Knoten „schnell“ erreicht werden kann.

  6. Genauer wird dies z. B. in [15] behandelt.

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Authors and Affiliations

  1. Hochschule RheinMain, Wiesbaden, Deutschland

    Michael Meyer & Steffen Reith

  2. Julius-Maximilians-Universität Würzburg, Würzburg, Deutschland

    Michael Meyer

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  1. Michael Meyer
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  2. Steffen Reith
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Correspondence to Michael Meyer.

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Meyer, M., Reith, S. Elliptische Kurven in der Post-Quantum-Kryptographie. Math Semesterber 66, 31–47 (2019). https://doi.org/10.1007/s00591-018-00239-8

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