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In the U.S. government’s ongoing campaign to protect data in the age of quantum computers, a powerful new attack that used a single traditional computer to completely break a fourth-round candidate highlights the risks involved in ‘standardization of the next generation of encryption algorithms.
Last month, the US Department of Commerce’s National Institute of Standards and Technology, or NIST, selected four post-quantum computing encryption algorithms to replace algorithms such as RSA, Diffie-Hellman, and Elliptic Curve Diffie-Hellman , which cannot withstand the attacks of a quantum computer.
In the same move, NIST advanced four additional algorithms as possible replacements pending further testing in the hope that one or more of them may also be suitable encryption alternatives in a post-quantum world. The new attack breaks SIKE, which is one of the last four additional algorithms. The attack has no impact on the four PQC algorithms selected by NIST as approved standards, all of which are based on completely different mathematical techniques than SIKE.
Totally getting SIKE
SIKE, short for Supersingular Isogeny Key Encapsulation, is probably out of order thanks to research published over the weekend by researchers from the Computer Security and Industrial Cryptography group at KU Leuven. The paper, titled An Efficient Key Recovery Attack on SIDH (Preliminary Version), describes a technique that uses complex mathematics and a single traditional computer to recover the encryption keys that protect SIKE-protected transactions. The whole process only takes about an hour.
“The newly discovered weakness is clearly a significant blow to SIKE,” David Jao, a professor at the University of Waterloo and co-inventor of SIKE, wrote in an email. “The attack is really unexpected.”
The advent of public-key encryption in the 1970s was a breakthrough because it allowed parties who had never met to securely trade encrypted material that an adversary could not crack. Public key encryption is based on asymmetric keys, with a private key used to decrypt messages and a separate public key for encryption. Users make their public key widely available. As long as your private key remains secret, the scheme remains secure.
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In practice, public key cryptography can often be unwieldy, so many systems rely on key encapsulation mechanisms, which allow parties who have never met before to jointly agree on a symmetric key in a public medium such as the Internet. Unlike symmetric key algorithms, the key encapsulation mechanisms used today are easily broken by quantum computers. SIKE, before the new attack, was thought to avoid these vulnerabilities by using a complex mathematical construct known as a supersingular isogeny graph.
The cornerstone of SIKE is a protocol called SIDH, short for Supersingular Isogeny Diffie-Hellman. The research paper released over the weekend shows how the IHRS is vulnerable to a theorem known as “glue-and-split” developed by mathematician Ernst Kani in 1997, as well as to tools devised by fellow mathematician Everett W. Howe , Franck Lepr’evost, and Bjorn Poonen in 2000. The new technique is based on what is known as an “adaptive GPS attack,” described in a 2016 paper. The math behind the latest attack is guaranteed to they are impenetrable to most non-mathematicians. Here is the closest possible:
“The attack takes advantage of the fact that SIDH has auxiliary points and that the degree of secret isogeny is known,” explained Steven Galbraith, professor of mathematics at the University of Auckland and the “G” of the GPST adaptive attack, in a brief write-up on the new attack. “SIDH auxiliary points have always been a nuisance and a potential weakness, and have been exploited for fault attacks, adaptive GPST attack, twist point attacks, etc.
It continued:
Let be the base curve and let have order. Let be such that there exists an isogeny of degree with , , and
A key aspect of SIDH is that it is not calculated directly, but as a composition of isogenies of degree 3. In other words, there is a sequence of curves connected by 3-isogenies.
Essentially, as in GPST, the attack determines the intermediate curves and thus ultimately determines the private key. In one step, the attack does a brute-force search of every possible one, and the magic ingredient is a gadget that shows which is the right one.
(The above is oversimplified, the isogenies in the attack are not of degree 3 but of degree a small power of 3.)
More important than understanding the math, Jonathan Katz, an IEEE member and professor in the computer science department at the University of Maryland, wrote in an email: “The attack is entirely classical and does not require quantum computers at all.” .