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22.b) DoublingDoubling points are obtained by multiplying each pointon a curve using the following method.P = (_ , _)2P = P+P =3P = 2P + P =4P = 3P + P =...nP =where n = p-1c) Generator PointGenerator points are obtained by considering theDoubling point which generates point (0, 0) at the lastresult of (P-1) and has no repetition of the Doublingpoint. For Example 1, the Generator points obtainedare:(8, 2) (8, 9) (9, 2) (9, 9)d) Encryption & DecryptionFor this part, let’s consider the Generator point to be(8,9). Therefore, the Doubling points are:1P = (8, 9)2P = (10, 6)3P = (9, 9)4P = (5, 2)5P = (1, 0)Secret Key, kA = 9Public Key, QA = kAP= 9P= (8, 2)plaintext, MA = (5,2)ciphertext, CA= MA + KAQBlog CA = log (5,2) + 9 log (9,2)= 4 + 9 (7)= 67 mod (p-1)= 67 mod 10= 7 CA = (9, 2) A sends QA and CA to BQA = (8, 2)CA = (9, 2)6P = (5, 9)7P = (9, 2)8P = (10, 5)9P = (8, 2)10P = (0, 0)Let’s assume Alice wants to send a secret messageto Bob. Both Alice and Bob have their own secret keyand public key. Diagram 1 shows the encryption anddecryption process using ECC.AliceEncryptQAQBSecret Key, kB = 7Public Key, QB = kBP= 7P= (9, 2) B receives QA and CA from AQA = (8, 2)CA = (9, 2)BobDecryptKBQA = 7 log (8, 2)= 7 (9) mod (p-1)= 63 mod 10= 3CA = MA + KAQB(9, 2) = MA + 7 log (8,2)log MA = log (9,2) - 7 log (8,2)= 7 – 3= 4 MA = (5, 2)Diagram 1: Encryption & Decryption process using ECC.ConclusionFrom the discussion above, we know that ECC is importantto public key cryptography algorithms and key exchangealgorithms because it is based on complex mathematicalconstructs. For more information about ECC, below are theavailable books that you can use as references1. http://www.cacr.math.uwaterloo.ca/ecc/ by DarrelHankerson, Alfred Menezes, and Scott Vanstone.2. http://www.hpl.hp.com/research/info_theory/ellipbook.html by Ian Blake, Gadiel Seroussi and NigelSmart.3. http://www.springer.com/math/numbers/book/978-0-387-77993-5 by J. Hoffstein, J. Pipher and J. H.Silverman.4. h t t p : / / w w w . s p r i n g e r o n l i n e . c o m / s g w / c d a /frontpage/0,11855,5-40109-22-33358649-0,00.htmlby Alfred Menezes. •References[1] Elliptic Curve Cryptography. http://en.wikipedia.org/wiki/ Elliptic_curve_cryptography[2] Public-Key Cryptosystem Based on Elliptic Curve ByJ.J. Botes and W.T. Penzhornhttp://eref.uqu.edu.sa/files/Others/Elliptic%20Curves/Public-key%20cryptosystems%20based%20on%20elliptic%20curves.pdf[3] Elliptic Curve Public Key Cryptosystem By Erik De Winand Bart Preneelh t t p : / / w w w . s p r i n g e r l i n k . c o m / c o n t e n t /wwjefcpuet8gtu83/fulltext.pdf[4] An Intro to Elliptical Curve Cryptographyh t t p : / / w w w . d e v i c e f o r g e . c o m / a r t i c l e s /AT4234154468.html