Advantage of ECC

1
Advantage of ECC
(1)The security of ECC is based on the discrete
logarithm problem of elliptic curves over
finite fields. (2)There are no sub-exponential
time algorithm to solve it. Thus 1024-bit RSA
is considered as secure as160-bit ECC. (3)ECC
is suitable for the implementation over memory
constraint computing environment like smart
cards.
2
Elliptic Curves
(Weierstrass-form of an elliptic curve)
p is a primegt3, discriminant
The set of points on the
curve and the point of infinity
has an additive group structure.
Addition (ECADD)
Doubling (ECDBL)
3
Standard Formula
(Weierstraß-form of an elliptic curve)
Standard Formula
4
Example
(Weierstraß-form of an elliptic curve)
a 1, b 6, p 11. E(1,6)/GF(11) O, (2,4),
(2,7), (3,5), (3,6), (5,2), (5,9),
(7,2), (7,9), (8,3), (8,8),
(10,2), (10,9)
ECADD (2,4) (10,9) (r2 - x1 - x2), r(x1 -
x3) - y1) (3,5) r (y1-y2)/(x1
- x2) (9-4)/(10-2) mod 11 2 ECDBL 2(8,8)
(r2 - (x1 x1), r(x1 - x3) - y1) (7,2)
r (3x12 1)/(2y1) mod 11 10
5
Scalar Multiplication
To compute
times
6 ECDBLs 2 ECADDs
99 ECADDs
Binary chain (from the MSB) Q0 P for in-2
down to 0 Q0 ECDBL(Q0) if ki1 then
Q0 ECADD(Q0,P) return(Q0)
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SECG (Standards for Efficient Cryptography Group)
secp160r1 p FFFFFFFF FFFFFFFF FFFFFFFF
FFFFFFFF 7FFFFFFF a FFFFFFFF FFFFFFFF FFFFFFFF
FFFFFFFF 7FFFFFFC b 1C97BEFC 54BD7A8B 65ACF89F
81D4D4AD C565FA45 x 4A96B568 8EF57328 46646989
68C38BB9 13CBFC82 y 23A62855 3168947D 59DCC912
04235137 7AC5FB32 n 01 00000000 00000000
0001F4C8 F927AED3 CA752257 h 01
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