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Elliptic Curve Cryptography
Elliptic curve parameters over the finite field Fp
T = (q, F R, a, b, G, n, h
q = the prime p
a,b! the curve coeffiecient
G! the base point (G",Gy
n! the order of G h! E(Fq #n$
%&' = "& ) a" ) b
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Elliptic Curve Cryptography (ECC
ECC depends on the hardness of the discretelogarithm problem
*et + and be t-o points on an elliptic curve
such that .+ = , -here . is a scalar$ Given +and , it is hard to compute .
. is the discrete logarithm of to the base +$
The main operation is point multiplication /ultiplication of scalar . 0 p to achieve another
point
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+oint 1ddition
+oint addition is the addition of t-o points 2 and3 on an elliptic curve to obtain another point *on the same elliptic curve$
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+oint 4oubling
+oint doubling is the addition of a point 2 on theelliptic curve to itself to obtain another point *on the same elliptic curve$
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+oint /ultiplication
.+=
+oint multiplication is achieved by point additionand point doubling
+oint addition, adding t-o points 2 and 3 toobtain another point * i$e$, * = 2 ) 3$
+oint doubling, adding a point 2 to itself to
obtain another point * i$e$ * = '2$
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+oint /ultiplication e"ample
*et . be a scalar that is multiplied -ith the point+ to obtain another point on the curve$ i$e$ tofind = .+$
5f . = ' then .+ = '$+ = '('('('+ ) + ) + )+
1s you can see point addition and pointdoubling are used to create
The above method is called 6double and add7method for point multiplication
8on91d:acent Form and -indo- 8on91d:acentForm are other methods
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Elliptic Curve 4igital ;ignature1lgorithm ;igning
For signing a message m by sender 1, using17s private .ey d
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Elliptic Curve 4igital ;ignature1lgorithm Derification
For to authenticate 1s signature, musthave 17s public .ey
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Elliptic Curve 4iffie ellman
a .ey pair consisting of a private .ey d (arandomly selected integer less than n, -here nis the order of the curve, an elliptic curve
domain parameter and a public .ey = d 0 G (G is the generator point,
an elliptic curve domain parameter$
*et (d1, 1 be the private .ey 9 public .ey pairof 1 and (d, be the private .ey 9 public.ey pair of
its not possible to obtain the shared secret for a
third party$
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Elliptic Curve 4iffie ellman +t$ '
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Reason For Ise
;maller .ey siJe
Faster than R;1
Good for handhelds and cell phones
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85;T Reccomend Curves
85;T reccomends p selections of
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Reference>
L#secB@ Certicom, ;tandards for Efficient Cryptography, ;EC '! Recommended Elliptic Curve
4omain +arameters, Dersion @ Mpenssl, http!##---$openssl$org
>H@ Certicom,http!##---$certicom$com#inde"$phpOaction=eccNtutorial,home >P@ 1lfred 2$ /eneJes, +aul C$ van Morschot and ;cott 1$ Danstone, andboo. of 1pplied