Shamus Software Limited

Shamus Standard Curves

These curves use the standard Weierstrass parameterisation, and are of the form:-

y² = x³ +Ax +B mod p

where p is a prime congruent to 3 mod 4, and A is fixed at -3. A quarter of all randomly generated curves can be transformed into this form.The former condition makes it easier to find points on the curve, and the latter make calculations on the curve somewhat faster.

The motivation is provide a set of curves which, within the limitations mentioned above, are otherwise in no way special. It is thought that by using such curves the user is safe against cryptanalytic advances, except in a circumstance where the whole premise behind Elliptic Curve cryptography collapses and a sub-exponential solution is found for the most general discrete logarithm problem in the elliptic curve setting.

Each curve is with respect to a prime p which is n bits in length. In each case the number of points q on the curve is itself a prime. The prime p is found as the first prime congruent to 3 mod 4 which is found by incrementing a number n bits in length, formed from the first n bits of the mathematical constant pi=3.141592.... The parameter B is formed from the first n bits of the mathematical constant e=2.71828...., incremented until q is prime.

ssc-160

n=160
B=993193335754933797118314178888153828594854512705
p=1147860701762054730346201299935827782113538756127
q=1147860701762054730346200648614608152209809891831

ssc-192

n=192
B=4265732895672588129268258440977714335632089762934383523494
p=4930024174431634640599033341057067222865862716297522433299
q=4930024174431634640599033341125441632693811654341940586403

ssc-224

n=224
B=183211832803851459388849904148752293363701930199395702272
  57813318147
p=211742925976732701691935620490537177918824237613235850561
  62680913631
q=211742925976732701691935620490537231344420991210242625510
  89688143309

ssc-256

n=256
B=786888830132762000916982485371625819202097623698479300223
  67595957783191893217
p=909428942229415810700587356944324656633483443320981074896
  93037779484723616779
q=909428942229415810700587356944324656632884146161715094318
  79910319924502217783

ssc-288

n=288
B=337966179100791213208996178567593129982221810838428315939
  365373128820605838874928979766
p=390596756491121423614434954606695289304724084762108334731
  724254341779347664665278286219
q=390596756491121423614434954606695289304724116479393090921
  502092797686514928150248753237

ssc-320

n=320
B=1451553686391976948456801799936788618707919738968947956999
  929796583121697128874465400872041660580
p=1677600295053042228788960243555000810201048522356787237681
  776606087928304667951345024875097229491
q=1677600295053042228788960243555000810201048522357873106251
  579120122685384485967275546948559607409

ssc-384

n=384
B=2677643936212245379258831955273195965014103242523976013961
  7629033244994517401871440317035340712170298670944533378961
p=3094626330082310195488842525978429610886059417792993623196
  1025381527827855583154673559277957637088071546809309873019
q=3094626330082310195488842525978429610886059417792993623195
  9195086011429040851460901626189237585847628753659044398489

ssc-512

n=512
B=9111550163858012281440901732746538838772262590143654133938
  6747435421078854920153908512486180420566799833852077056256
  99101049041930943171450852516780927629
p=1053046772336265905486170537113984702631399932837231365139
  8671272025951445569024729948471343061931586610942824229083
  371331823229156399790385588443550959087
q=1053046772336265905486170537113984702631399932837231365139
  8671272025951445569144524507377363887941433449823713742916
  287342504795006316114468040283111710577

These curves may be used freely without restriction.

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