of time they take to decrypt; and software vulnerabilities leading to stack overflows and other hacks. If you're implementing public-key cryptography you need to consult up-to-date standards, use properly accredited toolkits, and get someone knowledgeable to evaluate what you've done. And please don't write the actual crypto code on your own – doing it properly requires a lot of different skills, from computational number theory to side-channel analysis and formal methods. Even using good crypto libraries gives you plenty of opportunities to shoot your foot off.
5.7.2.3 ElGamal digital signature and DSA
Suppose that the base
So
or
An ElGamal signature on the message
A few more details need to be fixed up to get a functional digital signature scheme. As before, bad choices of
DSA assumes a prime
The hash function used by default is SHA2568.
DSA is the classic example of a randomised digital signature scheme without message recovery. The most commonly-used version nowadays is ECDSA, a variant based on elliptic curves, which we'll discuss now – this is for example the standard for cryptocurrency and increasingly also for certificates in bank smartcards.
5.7.3 Elliptic curve cryptography
Discrete logarithms and their analogues exist in many other mathematical structures. Elliptic curve cryptography uses discrete logarithms on an elliptic curve – a curve given by an equation like
First is performance; they give versions of the familiar primitives such as Diffie-Hellmann key exchange and the Digital Signature Algorithm that use less computation, and also have shorter variables; both are welcome in constrained environments. Elliptic curve cryptography is used in applications from the latest versions of EMV payment cards to Bitcoin.
Second, some elliptic curves have a bilinear pairing which Dan Boneh and Matt Franklin
