โ–„โ–„โ–„ย .ย โ–„โ–„ยทย ยทโ–„โ–„โ–„โ–„ย ย .โ–„โ–„ย ยทย ย โ–„โ–„โ–„ยทย 
โ–€โ–„.โ–€ยทโ–โ–ˆย โ–Œโ–ชโ–ˆโ–ˆโ–ชย โ–ˆโ–ˆย โ–โ–ˆย โ–€.ย โ–โ–ˆย โ–€โ–ˆย 
โ–โ–€โ–€โ–ชโ–„โ–ˆโ–ˆย โ–„โ–„โ–โ–ˆยทย โ–โ–ˆโ–Œโ–„โ–€โ–€โ–€โ–ˆโ–„โ–„โ–ˆโ–€โ–€โ–ˆย 
โ–โ–ˆโ–„โ–„โ–Œโ–โ–ˆโ–ˆโ–ˆโ–Œโ–ˆโ–ˆ.ย โ–ˆโ–ˆย โ–โ–ˆโ–„โ–ชโ–โ–ˆโ–โ–ˆย โ–ชโ–โ–Œ
ย โ–€โ–€โ–€ย ยทโ–€โ–€โ–€ย โ–€โ–€โ–€โ–€โ–€โ€ขย ย โ–€โ–€โ–€โ–€ย ย โ–€ย ย โ–€ย 
.โ–„โ–„ย ยทย ย โ–„โ–„โ–„ยทย ย โ–ย โ–„ย ยทโ–„โ–„โ–„โ–„ย ย โ–„โ–„โ–„โ–„ยทย ย ย ย ย ย ย โ–โ–„โ€ขย โ–„ย 
โ–โ–ˆย โ–€.ย โ–โ–ˆย โ–€โ–ˆย โ€ขโ–ˆโ–Œโ–โ–ˆโ–ˆโ–ˆโ–ชย โ–ˆโ–ˆย โ–โ–ˆย โ–€โ–ˆโ–ชโ–ชย ย ย ย ย ย โ–ˆโ–Œโ–ˆโ–Œโ–ช
โ–„โ–€โ–€โ–€โ–ˆโ–„โ–„โ–ˆโ–€โ–€โ–ˆย โ–โ–ˆโ–โ–โ–Œโ–โ–ˆยทย โ–โ–ˆโ–Œโ–โ–ˆโ–€โ–€โ–ˆโ–„ย โ–„โ–ˆโ–€โ–„ย ย ยทโ–ˆโ–ˆยทย 
โ–โ–ˆโ–„โ–ชโ–โ–ˆโ–โ–ˆย โ–ชโ–โ–Œโ–ˆโ–ˆโ–โ–ˆโ–Œโ–ˆโ–ˆ.ย โ–ˆโ–ˆย โ–ˆโ–ˆโ–„โ–ชโ–โ–ˆโ–โ–ˆโ–Œ.โ–โ–Œโ–ชโ–โ–ˆยทโ–ˆโ–Œ
ย โ–€โ–€โ–€โ–€ย ย โ–€ย ย โ–€ย โ–€โ–€ย โ–ˆโ–ชโ–€โ–€โ–€โ–€โ–€โ€ขย ยทโ–€โ–€โ–€โ–€ย ย โ–€โ–ˆโ–„โ–€โ–ชโ€ขโ–€โ–€ย โ–€โ–€
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yยฒ= xยณ+ax+b / yยฒโ‰ก xยณ+ax+b(mod N)
  • Field F=โ„ / F=โ„คโ‚š(p: prime number > 2)
  • ฮ” = -16(4aยฒ + 27bยฒ) โ‰  0
  • a, b โˆˆ F
  • O: Point at infinity
  • E(F) = {(x,y) โˆˆ F ร— F : yยฒ = xยณ + ax + b} โˆช {O}

Real Number Curve

Visualizes the curve over real numbers with both positive and negative coordinates

Zโ‚š Modular Curve (secp256r1 (simplified))

Shows discrete points in finite field Zโ‚š where coordinates range from 0 to p-1

Real Gy: ยฑ6.557
Mod Gy: No value

Note: These are simplified values for visualization. The modular curve shows points in Zโ‚š (integers mod p) where both x and y range from 0 to p-1. The actual curve parameters are much larger.

Curve: NIST P-256 curve used in TLS/SSL

secp256r1 a: 0xFFFFFFFF00000001000000000000000000000000FFFFFFFFFFFFFFFFFFFFFFFC

secp256r1 b: 0x5AC635D8AA3A93E7B3EBBD55769886BC651D06B0CC53B0F63BCE3C3E27D2604B

secp256r1 p: 0xFFFFFFFF00000001000000000000000000000000FFFFFFFFFFFFFFFFFFFFFFFF

TODO:
TODO:
  1. Reference 1
  2. Reference 2
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