View source for Secp256k1

[[File:Elliptic-Curve-E-0-7-Real-Number.png|thumb |This is a graph of secp256k1's elliptic curve ''y<sup>2</sup> = x<sup>3</sup> + 7'' over the real numbers. Note that because secp256k1 is actually defined over the field Z<sub>p</sub>, its graph will in reality look like random scattered points, not anything like this.]]

'''secp256k1''' refers to the parameters of the elliptic curve used in Bitcoin's public-key [[cryptography]], and is defined in ''Standards for Efficient Cryptography (SEC)'' (Certicom Research, http://www.secg.org/sec2-v2.pdf). Currently Bitcoin uses secp256k1 with the [[Elliptic Curve Digital Signature Algorithm|ECDSA]] algorithm.

secp256k1 was almost never used before Bitcoin became popular, but it is now gaining in popularity due to its several nice properties. Most commonly-used curves have a random structure, but secp256k1 was constructed in a special non-random way which allows for especially efficient computation. As a result, it is often more than 30% faster than other curves if the implementation is sufficiently optimized. Also, unlike the popular NIST curves, secp256k1's constants were selected in a predictable way, which significantly reduces the possibility that the curve's creator inserted any sort of backdoor into the curve.

=== Technical details ===

As excerpted from ''Standards'': 

The elliptic curve domain parameters over F''<sub>p</sub>'' associated with a Koblitz curve secp256k1 are specified
by the sextuple T = (''p,a,b,G,n,h'') where the finite field F''<sub>p</sub>'' is defined by:
* ''p'' = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE FFFFFC2F
* = 2<sup>256</sup> - 2<sup>32</sup> - 2<sup>9</sup> - 2<sup>8</sup> - 2<sup>7</sup> - 2<sup>6</sup> - 2<sup>4</sup> - 1

The curve ''E'': ''y<sup>2</sup> = x<sup>3</sup>+ax+b'' over F''<sub>p</sub>'' is defined by:
* ''a'' = 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000000
* ''b'' = 00000000 00000000 00000000 00000000 00000000 00000000 00000000 00000007

The [[base point G]] in compressed form is:
* ''G'' = 02 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B 16F81798
and in uncompressed form is:
* ''G'' = 04 79BE667E F9DCBBAC 55A06295 CE870B07 029BFCDB 2DCE28D9 59F2815B 16F81798 483ADA77 26A3C465 5DA4FBFC 0E1108A8 FD17B448 A6855419 9C47D08F FB10D4B8
Finally the order ''n'' of ''G'' and the cofactor are:
* ''n'' = FFFFFFFF FFFFFFFF FFFFFFFF FFFFFFFE BAAEDCE6 AF48A03B BFD25E8C D0364141
* ''h'' = 01

=== Properties ===

* secp256k1 has characteristic ''p'', it is defined over the prime field ℤ<sub>''p''</sub>. Some other curves in common use have characteristic ''2'', and are defined over a binary Galois field ''GF(2<sup>n</sup>)'', but secp256k1 is not one of them.
* As the ''a'' constant is zero, the ''ax'' term  in the curve equation is always zero, hence the curve equation becomes ''y<sup>2</sup> = x<sup>3</sup> + 7''.

==Attribution==
This content is based on content sourced from https://en.bitcoin.it/wiki/Secp256k1 under [https://creativecommons.org/licenses/by/3.0/ Creative Commons Attribution 3.0]. Although it may have been extensively revised and updated we acknowledge the original authors.