A few days ago I blogged about the elliptic curve secp256k1 and its use in Bitcoin. This curve has a sibling, secp256r1. Note the “r” in the penultimate position rather than a “k”. Both are defined in SEC 2: Recommended Elliptic Curve Domain Parameters. Both are elliptic curves over a field zp where p is a 256-bit prime (though different primes for each curve).
The “k” in sepc256k1 stands for Koblitz and the “r” in sepc256r1 stands for random. A Koblitz elliptic curve has some special properties that make it possible to implement the group operation more efficiently. It is believed that there is a small security trade-off, that more “randomly” selected parameters are more secure. However, some people suspect that the random coefficients may have been selected to provide a back door.
Both elliptic curves are of the form y² = x³ + ax + b. In the Koblitz curve, we have
a = 0 b = 7
and in the random case we have
a = FFFFFFFF 00000001 00000000 00000000 00000000 FFFFFFFF FFFFFFFF FFFFFFFC b = 5AC635D8 AA3A93E7 B3EBBD55 769886BC 651D06B0 CC53B0F6 3BCE3C3E 27D2604B
You can find the rest of the elliptic curve parameters in the SEC 2 report. For some help understanding what the parameters mean and how to decode them, see my earlier post.
The NSA recommends the random curve for government use. It is also known as NIST P-256. Or rather it did recommend P-256 as part of its Suite B of cryptography recommendations. In August 21015 the NSA announced its concern that in the future, quantum computing could render the Suite B methods insecure. As far as we know, quantum computing at scale is years, maybe decades, away. But it takes a long time to develop quality encryption methods, and so the NSA and NIST are urging people to think ahead. (Update: The NSA recommends P-384 until post quantum methods mature.)
Bitcoin chose to use the less popular Koblitz curve for the reasons mentioned above, namely efficiency and concerns over a possible back door in the random curve. Before Bitcoin, secp256k1 was not widely used.
Related post: RSA numbers and factoring