Elliptic Curve The elliptic curve cryptosystem are modifications of other systems that work in the domains of elliptic curve rather than finite fields The elliptic curve cryptosystems appear to remain secure for smaller keys than other public-key cryptosystems The use of elliptic curves was first proposed by Miller(1986) and Koblitz(1987) Already, eCc is showing up in standardization efforts, including IEEE P1363 Standard for Public-Key Cryptography 16
16 Elliptic Curve • The elliptic curve cryptosystem are modifications of other systems that work in the domains of elliptic curve rather than finite fields. • The elliptic curve cryptosystems appear to remain secure for smaller keys than other public-key cryptosystems. • The use of elliptic curves was first proposed by Miller (1986) and Koblitz (1987). • Already, ECC is showing up in standardization efforts, including IEEE P1363 Standard for Public-Key Cryptography
RSA Modular Exponentiation y x moa n 360 RSA key size Processing 300 k[bits] time t[s] 512 240 768 22 180 1024 48 120 1536 150 60 2048 335 5127681024 1536 2048 RSa key size k [bits 17
17 RSA Modular Exponentiation y = x e mod n 512 768 1024 1536 2048 0 60 120 180 240 300 360 RSA key size k [bits] processing time t [s] RSA key size k [bits] Processing time t [s] 512 8 768 22 1024 48 1536 150 2048 335
What are Elliptic curves? General form y=xt ax+ b Condition for distinct 1 single roots. 4a3+27b2≠0 Example 4 x(x-2)(x+2) 0
18 -3 -2 -1 0 1 2 3 -4 -3 -2 -1 0 1 2 3 4 y 2 = x 3 + ax + b 4a 3 + 27b 2 0 General form: Condition for distinct single roots: Example: y 2 = x 3 − 4x = x(x −2)(x +2) What are Elliptic Curves?