The use of point pairing becomes very useful when many people are required to sign one document. This is typical in a contract situation when several people are agreeing to a set of requirements. If we used the method described in the blog on signatures, each person would sign the document, and then the verification process would require checking every single signature. By using...]]>

When my kids play games they talk about "leveling up". To get to the mathematics of pairing points over elliptic curves, we need to "level up" to extension fields from prime fields. If you've done any work with binary codes this is actually very familiar. Rather than base 2 fields, we are going to use base $p$ fields. An extension field is over $F_{p^k}$. It is...]]>

To recap the basic math, an elliptic curve over a finite field has points $(x, y)$ which satisfy the equation $$y^2 = x^3 + a x + b \text{ mod } q.$$When points are added to themselves multiple times we write the multiplication as $$Y = k P$$where $k$ is an integer. Since the number of points is finite, after a while we get to a value of $k = n$ such that $$\mathscr O = n P....]]>

Cryptographic security has a lot of components. The simple stuff is the mathematics which is what I want to talk about. The hard stuff is preventing people from giving away things that should be secret (Loose lips sink ships still holds today!). The cryptographic security I want to talk about here comes from solving a mathematical problem to find a secret. The assumption is the...]]>

Cryptography is the art of hiding messages, NOT writing on graves, which is a direct translation a friend of mine once asked. I should have said "that's engraving!", but I was a week late. The main engine of encrypting a message uses a single key and a fast algorithm. The NIST standard is AES which can use key sizes of 128 bits, 192 bits, or 256 bits. Each of these is considered a...]]>

Last year I was asked by Manning Publications if I wanted to write another book on elliptic curve crypto. I said that as long as I can learn a lot of new math I'd love to. So I spent 6 months learning math and then another year writing. The first three chapters are now online here: http://mng.bz/D9NA

Along the way I had proposed to explain an encryption scheme described on NIST...]]>

The mathematics of number theory and elliptic curves can take a life time to learn because they are very deep subjects. As engineers we don't have time to earn PhD's in math along with all the things we have to learn just to make communications systems work. However, a little learning can go a long way to helping make our communications systems secure - we don't need to know...]]>