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Elliptic Curve

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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