NIST Koblitz Curves Parameters

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FIPS 186-2 by NIST

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The parameter sets of the five binary koblitz curves standardized by NIST are listed below. The curves are of the form y^2 + xy = x^3 + ax^2 + 1 over a binary field. For the five curves the following parameters are listed:

1. p(t): the reduction polynomial (in explicit and hexadecimal form)
2. a: the curve's a coefficient
3. G_x, G_y: the x and y coordinates of the base point G
4. n: the base point's order
5. h: the curve's cofactor

Contents 1 K163 2 K233 3 K283 4 K409 5 K571

[edit] K163

p(t) = t^163 + t^7 + t^6 + t^3 + 1 
     = 800000000000000000000000000000000000000C9
a    = 1
G_x  = 2fe13c0537bbc11acaa07d793de4e6d5e5c94eee8
G_y  = 289070fb05d38ff58321f2e800536d538ccdaa3d9
n    = 5846006549323611672814741753598448348329118574063
h    = 2

[edit] K233

p(t) = t^233 + t^74 + 1
     = 20000000000000000000000000000000000000004000000000000000001
a    = 0
G_x  = 17232ba853a7e731af129f22ff4149563a419c26bf50a4c9d6eefad6126
G_y  = 1db537dece819b7f70f555a67c427a8cd9bf18aeb9b56e0c11056fae6a3
n    = 3450873173395281893717377931138512760570940988862252126328087024741343
h    = 4

[edit] K283

p(t) = t^283 + t^12 + t^7 + t^5 + 1
     = 800000000000000000000000000000000000000000000000000000000000000000010A1
a    = 0
G_x  = 503213f78ca44883f1a3b8162f188e553cd265f23c1567a16876913b0c2ac2458492836
G_y  = 1ccda380f1c9e318d90f95d07e5426fe87e45c0e8184698e45962364e34116177dd2259
n    = 3885337784451458141838923813647037813284811733793061324295874997529815829704422603873
h    = 4

[edit] K409

p(t) = t^409 + t^87 + 1
     = 2000000000000000000000000000000000000000000000000000000000000000000000000000000008000000000000000000001
a    = 0
G_x  = 060f05f658f49c1ad3ab1890f7184210efd0987e307c84c27accfb8f9f67cc2c460189eb5aaaa62ee222eb1b35540cfe9023746
G_y  = 1e369050b7c4e42acba1dacbf04299c3460782f918ea427e6325165e9ea10e3da5f6c42e9c55215aa9ca27a5863ec48d8e0286b
n    = 330527984395124299475957654016385519914202341482140609642324395022880711289249191050673258457777458014096366590617731358671
h    = 4

[edit] K571

p(t) = t^571 + t^10 + t^5 + t^2 + 1
     = 80000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000425
a    = 0
G_x  = 26eb7a859923fbc82189631f8103fe4ac9ca2970012d5d46024804801841ca44370958493b205e647da304db4ceb08cbbd1ba39494776fb988b47174dca88c7e2945283a01c8972
G_y  = 349dc807f4fbf374f4aeade3bca95314dd58cec9f307a54ffc61efc006d8a2c9d4979c0ac44aea74fbebbb9f772aedcb620b01a7ba7af1b320430c8591984f601cd4c143ef1c7a3
n    = 1932268761508629172347675945465993672149463664853217499328617625725759571144780212268133978522706711834706712800825351461273674974066617311929682421617092503555733685276673
h    = 4