KGC_TEST/KGCAPP/3rdparty/miracl/docs/miracl-explained/miracl-standard-curves.md

• What Is Miracl
• Security Advisory
• Benchmarks
• Miracl Standard Curves
• IEEE 1363
• Elliptic Curves
• Licensing
• Reference Manual
  • Low Level Routines
  • Advanced Arithmetic Routines
  • Montgomery Arithmetic Routines
  • ZZn2 Arithmetic Routines
  • Encryption Routines
  • Elliptic Curve Routines
  • Floating Slash Routines
  • Structure Reference

MIRACL 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=18321183280385145938884990414875229336370193019939570227257813318147
p=21174292597673270169193562049053717791882423761323585056162680913631
q=21174292597673270169193562049053723134442099121024262551089688143309

ssc-256

n=256
B=78688883013276200091698248537162581920209762369847930022367595957783191893217
p=90942894222941581070058735694432465663348344332098107489693037779484723616779
q=90942894222941581070058735694432465663288414616171509431879910319924502217783

ssc-288

n=288
B=337966179100791213208996178567593129982221810838428315939365373128820605838874928979766
p=390596756491121423614434954606695289304724084762108334731724254341779347664665278286219
q=390596756491121423614434954606695289304724116479393090921502092797686514928150248753237

ssc-320

n=320
B=1451553686391976948456801799936788618707919738968947956999929796583121697128874465400872041660580
p=1677600295053042228788960243555000810201048522356787237681776606087928304667951345024875097229491
q=1677600295053042228788960243555000810201048522357873106251579120122685384485967275546948559607409

ssc-384

n=384
B=26776439362122453792588319552731959650141032425239760139617629033244994517401871440317035340712170298670944533378961
p=30946263300823101954888425259784296108860594177929936231961025381527827855583154673559277957637088071546809309873019
q=30946263300823101954888425259784296108860594177929936231959195086011429040851460901626189237585847628753659044398489

ssc-512

n=512
B=9111550163858012281440901732746538838772262590143654133938674743542107885492015390851248618042056679983385207705625699101049041930943171450852516780927629
p=10530467723362659054861705371139847026313999328372313651398671272025951445569024729948471343061931586610942824229083371331823229156399790385588443550959087
q=10530467723362659054861705371139847026313999328372313651398671272025951445569144524507377363887941433449823713742916287342504795006316114468040283111710577

These curves may be used freely without restriction.