Isogeny-based cryptography

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Elliptic curve cryptography is one of the most efficient, secure, and well-studied technologies available today for achieving public-key encryption and authentication, and enjoys widespread use in internet and software applications. Although these systems are generally believed to be secure today, we have known since 1994 that a universal quantum computer would be able to break most of these systems. However, one type of elliptic curve cryptography, called isogeny-based cryptography, may possibly be more resistant against attacks from quantum computers, because the underlying mathematical problems are more difficult to solve on a quantum computer. In 2011, I invented Supersingular Isogeny Diffie-Hellman (SIDH), the first practical public-key cryptosystem using isogenies. Since then, the field of isogeny-based cryptography has blossomed into one of the major branches of post-quantum cryptography (the study of cryptosystems resistant to quantum computer attacks). In 2017, the US government's National Institute of Standards and Technology (NIST) launched a public standardization process for post-quantum cryptosystems, and I along with a team of researchers submitted Supersingular Isogeny Key Encapsulation (SIKE) as a candidate for standardization. Currently SIKE has advanced to round 3 as an alternate candidate, and as quoted in the evaluation report: "NIST sees SIKE as a strong candidate for future standardization with continued improvements." The main drawback of SIKE and other isogeny-based cryptosystems is that the speed of isogeny computations is slower than alternative schemes, although it remains quite practical for many applications. In this proposal, we plan to pursue the development of faster isogeny cryptography implementations, more efficient methods of further reducing the size of public keys in isogeny cryptosystems, new models and techniques for analyzing the security of implementations of isogeny-based cryptosystems, and new isogeny-based cryptographic protocols supporting advanced functionality such as blockchains. These results will help to establish Canada at the forefront of the development of security and privacy technologies in the post-quantum era. Expected outcomes and impact of this work include the public standardization and widespread acceptance of isogeny-based cryptography, clear leadership of isogeny-based cryptography in size and space-constrained applications requiring small public keys, and modern-day functionality such as secure messaging and blockchain that remains secure against attacks from quantum computers. Students who work on this project will receive training in cryptography, computer security, software and hardware implementation, and mathematics, gaining valuable skills which will allow them to participate fully in the technology sector and in academic and industrial research.