1. Mathematics Towards Elliptic Curve Cryptography by Dr. R. Srinivasan Dean R & D and Post Graduate Studies RNS Institute of Technology, Bangalore Comp Sc. Dept, Mysore 10.9..2011

2. Cryptography Definitions 1 . Cryptography (or cryptology ; from Greek κρυπτός, kryptos , "hidden, secret "; and γράφειν , graphein , "writing", or -λογία , -logia , "study", respectively) [1] is the practice and study of hiding information . Modern cryptography intersects the disciplines of mathematics, computer science , and electrical engineering . 2 . Cryptography is the science of information security . The word is derived from the Greek kryptos , meaning hidden. Cryptography is closely related to the disciplines of cryptology and cryptanalysis 3. Discipline or techniques employed in protecting integrity or secrecy of electronic messages by converting them into unreadable (cipher text) form. Only the use of a secret key can convert the cipher text back into human readable (clear text) form. Cryptography software and/or hardware devices use mathematical formulas (algorithms) to change text from one form to another. Source: Internet

4. Hieroglyphs Hieroglyphs showing the words for Father, Mother, Son, Egyptian Hieroglyphs for Kids ! Source: Internet

5. Zimmermann’s Telegram – January 16, 1917 The message came as a coded telegram dispatched by the Foreign Secretary of the German Empire, ARTHUR ZIMMERMANN, on January 16, 1917, to the German ambassador in Washington D.C., Johann von Bernstorff, at the height ofWorld War I. On January 19, Bernstorff, per Zimmermann's request, forwarded the telegram to the German ambassador in Mexico, Heinrich von Eckardt. Source: Internet

6. Source: Internet

11. RS-RNSIT Whitfield Diffie Martin Hellman Pioneers of Public – Key Cryptography

14. Large Key Size RS-RNSIT Y = KX , Y- encrypted message of Plain Text Message “x” with Key K X = K -1 Y – Inverse operation must be difficult – larger the key more difficult

27. If p≠2 Weierstrass equation can be simplified by transformation to get the equation for some constants d,e,f and if p≠3 by transformation to get equation ELIPTIC CURVES - GENERALITY An elliptic curve over where p is a prime is the set of points (x,y) satisfying so-called Weierstrass equation for some constants u,v,a,b,c together with a single element 0 , called the point of infinity.

30. Real Elliptic Curve Examples RS-RNSIT a = - 4 and b = 0.7

34. Example of an Elliptic Curve Group over Fp (contd.) RS-RNSIT

37. Finite fields of the form GF 2 n (contd.) RS-RNSIT n  {113, 131, 163, 193, 233, 239, 283, 409, 571} Ref: Secg-talk@lists.certicom.com Field Reduction Polynomials F 2 113 f(x) = x 113 + x 9 + 1 F 2 131 f(x) = x 131 + x 8 + x 3 + x 2 + 1 F 2 163 f(x) = x 163 + x 7 + x 6 + x 3 +1 F 2 193 f(x) = x 193 + x 15 + 1 F 2 233 f(x) = x 233 + x 74 + 1 F 2 239 f(x) = x 239 + x 36 + 1 F 2 283 f(x) = x 283 + x 12 + x 7 + x 5 +1 F 2 409 f(x) = x 409 + x 87 + 1

39. Elliptic Curve Groups over F 2 n (contd.) RS-RNSIT Ex. g 5 = (g 4 )(g) = (g+1)g = g 2 + g = 0110 g 6 = g 4 .g 2 = (g+1)g 2 = g 3 +g 2 = 1100 g0 = 0001 g4 = 0011 g8 = 0101 g12 = 1111 g1 = 0010 g5 = 0110 g9 = 1010 g13 = 1101 g2 = 0100 g6 = 1100 g10 = 0111 g14 = 1001 g3 = 1000 g7 = 1011 g11 = 1110 g15 = 0001

41. Elliptic Curve Groups over F 2 n (contd.) RS-RNSIT

42. Adding Points P + Q on E - - P Q P+Q R

43. Doubling a Point P on E - - P 2*P R Tangent Line to E at P

44. Vertical Lines and an Extra Point at Infinity Add an extra point O “at infinity.” The point O lies on every vertical line. - - Vertical lines have no third intersection point Q O P Q = –P

46. A Numerical Example Using the tangent line construction, we find that 2P = P + P = (-7/4, -27/8). Using the secant line construction, we find that 3P = P + P + P = (553/121, -11950/1331) Similarly, 4P = (45313/11664, 8655103/1259712). As you can see, the coordinates become complicated. - - E : Y 2 = X 3 – 5X + 8 The point P = (1,2) is on the curve E.

58. Advantages of ECC Hence, ECC offers equivalent security with much small key size. Practical advantages of ECC : 1 Faster 2 Low power consumption 3 Low memory usage 4 Low CPU utilization 5 Benefits of over its competitors increases with increase in the security needs.

62. RS-RNSIT Thank You !