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1. Reductions of abelian surfaces over global function fields

   https://doi.org/10.1112/S0010437X22007473  

   Maulik, Davesh; Shankar, Ananth N.; Tang, Yunqing (April 2022, Compositio Mathematica)

   Let $$A$$ be a non-isotrivial ordinary abelian surface over a global function field of characteristic $p>0$ with good reduction everywhere. Suppose that $$A$$ does not have real multiplication by any real quadratic field with discriminant a multiple of $$p$$ . We prove that there are infinitely many places modulo which $$A$$ is isogenous to the product of two elliptic curves. 

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2. Exceptional jumps of Picard ranks of reductions of K3 surfaces over number fields

   https://doi.org/10.1017/fmp.2022.14  

   Shankar, Ananth N.; Shankar, Arul; Tang, Yunqing; Tayou, Salim (January 2022, Forum of Mathematics, Pi)

   Abstract Given a K3 surface X over a number field K with potentially good reduction everywhere, we prove that the set of primes of K where the geometric Picard rank jumps is infinite. As a corollary, we prove that either $$X_{\overline {K}}$$ has infinitely many rational curves or X has infinitely many unirational specialisations. Our result on Picard ranks is a special case of more general results on exceptional classes for K3 type motives associated to GSpin Shimura varieties. These general results have several other applications. For instance, we prove that an abelian surface over a number field K with potentially good reduction everywhere is isogenous to a product of elliptic curves modulo infinitely many primes of K . 

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3. Dronecrypt - An Efficient Cryptographic Framework for Small Aerial Drones

   https://doi.org/10.1109/MILCOM.2018.8599784  

   Ozmen, Muslum O.; Yavuz, Attila A. (October 2018, IEEE)

   Aerial drones are becoming an integral part of application domains including but not limited to, military operations, package delivery, construction, monitoring and search/rescue operations. It is critical to ensure the cyber security of networked aerial drone systems in these applications. Standard cryptographic services can be deployed to provide basic security services; however, they have been shown to be inefficient in terms of energy and time consumption, especially for small aerial drones with resource-limited processors. Therefore, there is a significant need for an efficient cryptographic framework that can meet the requirements of small aerial drones. We propose an improved cryptographic framework for small aerial drones, which offers significant energy efficiency and speed advantages over standard cryptographic techniques. (i) We create (to the best of our knowledge) the first optimized public key infrastructure (PKI) based framework for small aerial drones, which provides energy efficient techniques by harnessing special precomputation methods and optimized elliptic curves. (ii) We also integrate recent light-weight symmetric primitives into our PKI techniques to provide a full-fledged cryptographic framework. (iii) We implemented standard counterparts and our proposed techniques on an actual small aerial drone (Crazyflie 2.0), and provided an in-depth energy analysis. Our experiments showed that our improved cryptographic framework achieves up to 35× lower energy consumption than its standard counterpart. 

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4. Doubly isogenous genus-2 curves with 𝐷₄-action

   https://doi.org/10.1090/mcom/3891  

   Arul, Vishal; Booher, Jeremy; Groen, Steven; Howe, Everett; Li, Wanlin; Matei, Vlad; Pries, Rachel; Springer, Caleb (January 2024, Mathematics of Computation)

   Abstract. We study the extent to which curves over finite fields are characterized by their zeta functions and the zeta functions of certain of their covers. Suppose C and C ′ are curves over a finite field K, with K-rational base points P and P ′ , and let D and D ′ be the pullbacks (via the Abel–Jacobi map) of the multiplication-by-2 maps on their Jacobians. We say that (C, P) and (C ′ , P ′ ) are doubly isogenous if Jac(C) and Jac(C ′ ) are isogenous over K and Jac(D) and Jac(D ′ ) are isogenous over K. For curves of genus 2 whose automorphism groups contain the dihedral group of order eight, we show that the number of pairs of doubly isogenous curves is larger than na¨ıve heuristics predict, and we provide an explanation for this phenomenon. 

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5. EdSIDH: Supersingular Isogeny Die-Hellman Key Exchange on Edwards Curves

   https://doi.org/10.1007/978-3-030-05072-6_8  

   Azarderakhsh, Reza; Lang, B Elena; Jao, David; Koziel, Brian (January 2018, International Conference on Security, Privacy, and Applied Cryptography Engineering SPACE 2018: Security, Privacy, and Applied Cryptography Engineering)

   Problems relating to the computation of isogenies between elliptic curves defined over finite fields have been studied for a long time. Isogenies on supersingular elliptic curves are a candidate for quantum-safe key exchange protocols because the best known classical and quantum algorithms for solving well-formed instances of the isogeny problem are exponential. We propose an implementation of supersingular isogeny Diffie-Hellman (SIDH) key exchange for complete Edwards curves. Our work is motivated by the use of Edwards curves to speed up many cryptographic protocols and improve security. Our work does not actually provide a faster implementation of SIDH, but the use of complete Edwards curves and their complete addition formulae provides security benefits against side-channel attacks. We provide run time complexity analysis and operation counts for the proposed key exchange based on Edwards curves along with comparisons to the Montgomery form. 

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