A crippling flaw affecting millions—and possibly hundreds of millions—of encryption keys used in some of the highest-stakes security settings is considerably easier to exploit than originally reported, cryptographers declared over the weekend. The assessment came as Estonia abruptly suspended 760,000 national ID cards used for voting, filing taxes, and encrypting sensitive documents.

The critical weakness allows attackers to calculate the private portion of any vulnerable key using nothing more than the corresponding public portion. Hackers can then use the private key to impersonate key owners, decrypt sensitive data, sneak malicious code into digitally signed software, and bypass protections that prevent accessing or tampering with stolen PCs.

[...] One of the scenarios Bernstein and Lange presented in Sunday's post is that serious attackers can further reduce costs by buying dedicated computer gear, possibly equipped with GPU, field programmable gate array, and application-specific integrated circuit chips, which are often better suited for the types of mathematical operations used in factorization attacks. The estimates provided by the original researchers were based on the cost of renting equipment, which isn't as cost-effective when factorizing large numbers of keys. They also noted that compromising just 10 percent of cards used in country-wide voting might be enough to tip an election.