Examples Of Elliptic Curve Cryptography

1421 Words6 Pages
Elliptical Curve Cryptography: Elliptic curve cryptography is one of the emerging techniques that stand as an alternative for conventional public key cryptography. Elliptic curve cryptography has several applications of which smart cards are also one among them. A smart card is nothing but a single chip that contains microprocessor components. Smart cards are mainly used for secured sign-on in big organizations. The security feature of smart card is provided by elliptic curve cryptography. Elliptic curve cryptography for smart cards can be implemented through several ways. Of them implementation using Galois Field is one of the very famous techniques. Elliptic curves are not ellipses, instead, they are cubic curves of the form y3 = x3 + ax…show more content…
The curve is represented as E(R). The following figure is an elliptic curve satisfying the equation y2 = x3 – 3x + 3 Some Definitions of the Elliptical curve: • Scalar Multiplication: Given an integer k and a point P on the elliptic curve, the elliptic scalar multiplication kP is the result of adding Point P to itself k times. • Order: Order of a point P on the elliptic curve is the smallest integer r such that rP = O. Further if c and d are integers, then cP = dP iff c  d (mod r). • Curve Order: The number of points on the elliptic curve is called its curve order and is denoted…show more content…
Step 1: Selects a random integer r in [1, n – 1] Step 2: Computes R = rG Step 3: Computes K = hrQB = (Kx, Ky), checks that K  O Step 4: Computes keys k1||k2 = KDF(Kx) where KDF is a key derivation function, which derives cryptographic keys from a shared secret Computes c = ENCk1(m) where m is the message to be sent and ENC a symmetric encryption algorithm Step 5: Compute t = MACk2(c) where MAC is message authentication code Step 6: Sends (R, c, t) to Bob To decrypt a cipher text, Bob performs the following steps Step 1: Perform a partial key validation on R (check if R  O, check if the coordinates of R are properly represented elements in Fq and check if R lies on the elliptic curve defined by a and b) Step 2: Computes KB = h.dB.R = (Kx, Ky ) , check K  O Step 3: Compute k1, k2 = KDF (Kx) Step 4: Verify that t = MACk2(c) Step 5: Computes m = We can see that K = KB, since K = h.r.QB = h.r.dB.G = h.dB.r.G = h.dB.R = KB ECC Digital Signature: Alice, with domain parameters D = (q, FR, a, b, G, n, h), public key Q and private key d, does the following steps to sign the message m Step 1: Selects a Random number k  [1, n – 1] Step 2: Computes Point kG = (x, y) and r = x mod n, if r = 0 then goto Step

More about Examples Of Elliptic Curve Cryptography

Open Document