Ellis Elliptic Curve Crypto

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    10-Oct-2015

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Elliptic Curve

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<ul><li><p>5/20/2018 Ellis Elliptic Curve Crypto</p><p> 1/13</p><p>Elliptic Curve Cryptography</p><p> Elliptic curve parameters over the finite field Fp</p><p> T = (q, F R, a, b, G, n, h</p><p> q = the prime p</p><p> a,b! the curve coeffiecient</p><p> G! the base point (G",Gy</p><p>n! the order of G h! E(Fq #n$</p><p> %&amp;' = "&amp; ) a" ) b</p></li><li><p>5/20/2018 Ellis Elliptic Curve Crypto</p><p> 2/13</p><p>Elliptic Curve Cryptography (ECC</p><p> ECC depends on the hardness of the discretelogarithm problem</p><p> *et + and be t-o points on an elliptic curve</p><p>such that .+ = , -here . is a scalar$ Given +and , it is hard to compute .</p><p> . is the discrete logarithm of to the base +$</p><p> The main operation is point multiplication /ultiplication of scalar . 0 p to achieve another</p><p>point </p></li><li><p>5/20/2018 Ellis Elliptic Curve Crypto</p><p> 3/13</p><p>+oint 1ddition</p><p> +oint addition is the addition of t-o points 2 and3 on an elliptic curve to obtain another point *on the same elliptic curve$</p></li><li><p>5/20/2018 Ellis Elliptic Curve Crypto</p><p> 4/13</p><p>+oint 4oubling</p><p> +oint doubling is the addition of a point 2 on theelliptic curve to itself to obtain another point *on the same elliptic curve$</p></li><li><p>5/20/2018 Ellis Elliptic Curve Crypto</p><p> 5/13</p><p>+oint /ultiplication</p><p> .+=</p><p> +oint multiplication is achieved by point additionand point doubling</p><p> +oint addition, adding t-o points 2 and 3 toobtain another point * i$e$, * = 2 ) 3$</p><p> +oint doubling, adding a point 2 to itself to</p><p>obtain another point * i$e$ * = '2$</p></li><li><p>5/20/2018 Ellis Elliptic Curve Crypto</p><p> 6/13</p><p>+oint /ultiplication e"ample</p><p> *et . be a scalar that is multiplied -ith the point+ to obtain another point on the curve$ i$e$ tofind = .+$</p><p> 5f . = ' then .+ = '$+ = '('('('+ ) + ) + )+</p><p> 1s you can see point addition and pointdoubling are used to create </p><p> The above method is called 6double and add7method for point multiplication</p><p> 8on91d:acent Form and -indo- 8on91d:acentForm are other methods</p></li><li><p>5/20/2018 Ellis Elliptic Curve Crypto</p><p> 7/13</p><p>Elliptic Curve 4igital ;ignature1lgorithm ;igning</p><p> For signing a message m by sender 1, using17s private .ey d</p></li><li><p>5/20/2018 Ellis Elliptic Curve Crypto</p><p> 8/13</p><p>Elliptic Curve 4igital ;ignature1lgorithm Derification</p><p> For to authenticate 1s signature, musthave 17s public .ey </p></li><li><p>5/20/2018 Ellis Elliptic Curve Crypto</p><p> 9/13</p><p>Elliptic Curve 4iffie ellman</p><p> 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</p><p>domain parameter and a public .ey = d 0 G (G is the generator point,</p><p>an elliptic curve domain parameter$</p><p> *et (d1, 1 be the private .ey 9 public .ey pairof 1 and (d, be the private .ey 9 public.ey pair of </p><p> its not possible to obtain the shared secret for a</p><p>third party$</p></li><li><p>5/20/2018 Ellis Elliptic Curve Crypto</p><p> 10/13</p><p>Elliptic Curve 4iffie ellman +t$ '</p></li><li><p>5/20/2018 Ellis Elliptic Curve Crypto</p><p> 11/13</p><p>Reason For Ise</p><p> ;maller .ey siJe</p><p> Faster than R;1</p><p> Good for handhelds and cell phones</p></li><li><p>5/20/2018 Ellis Elliptic Curve Crypto</p><p> 12/13</p><p>85;T Reccomend Curves</p><p> 85;T reccomends p selections of</p></li><li><p>5/20/2018 Ellis Elliptic Curve Crypto</p><p> 13/13</p><p>Reference&gt;</p><p> L#secB@ Certicom, ;tandards for Efficient Cryptography, ;EC '! Recommended Elliptic Curve</p><p> 4omain +arameters, Dersion @ Mpenssl, http!##---$openssl$org</p><p>&gt;H@ Certicom,http!##---$certicom$com#inde"$phpOaction=eccNtutorial,home &gt;P@ 1lfred 2$ /eneJes, +aul C$ van Morschot and ;cott 1$ Danstone, andboo. of 1pplied</p></li></ul>