Abstract—The need for secure communication arises to protect
valuable information as technology becomes faster and more
efficient. We use cryptography, the art of secret writing, for adding
security to our communication. Most of the security architecture
uses public key cryptosystems for authentication and to secure the
communication. This paper represents the basic idea of elliptic
curve cryptography (ECC)-the emerging public key cryptographic
technique.
Keywords—Elliptic Curve Cryptography (ECC), Public-key
cryptography, Discrete Logarithm.
ECC is a kind of public key cryptosystem and was
introduced in the mid-1980s independently by Neil Koblitz
and Victor Miller. Main problem of conventional public key
cryptographic system is
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a, b = coefficients
However for finite fields a modified equation is used:
y 2 mod p = (x 3 +ax + b) mod p
where p is a prime number.
For the polynomial, (x 3 +ax + b) mod p, the discriminant
can be given as:
D = (4a 3 + 27b 2 ) mod p
This discriminant must not become zero for an elliptic
curve polynomial x 3 +ax + b to possess three distinct roots. If
the discriminant is zero, that would imply that two or more
roots have coalesced, giving the curves in singular form. It is
not safe to use singular curves for cryptography as they are
easy to crack. Due to this reason we generally take non-
singular curves for data encryption.
B. Elliptic Curve Basic Operations
The basic operations on elliptic curves are point addition
and point doubling. A scalar multiplication with a point can be
represented as a combination of addition operations.
Point Addition:
Adding two points P and Q using elliptic curve equation to
obtain another point R, i.e., R = P + Q.
Geometric explanation to point addition:
Suppose that P and Q are two distinct points on an elliptic
curve, and P is not -Q. To add the points P and Q, a line
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ELLIPTIC CURVE CRYPTOGRAPHY
Point Doubling:
Adding a point P to itself using elliptic curve equation to
obtain another point R, i.e. R = 2P.
Geometric explanation to point doubling:
To add a point P to itself, a tangent line to the curve is
drawn at the point P. The tangent line intersects the elliptic
curve at exactly one other point; -R. -R is reflected in the x-
axis to R. This operation is called doubling the point P; the
law for doubling a point on an elliptic curve group is defined
by: P + P = 2P = R.
Elliptic Curve Point Addition and Doubling Formulas:
Suppose P(x 1 ,y 1 ) and Q(x 2 ,y 2 ) are points on elliptic curve.
If x 1 =x 2 and y 1 =-y 2 , then P+Q= θ (point at infinity)
Otherwise P+Q=(x 3 ,y 3 )
where
x 3 = s 2 −x 1 −x 2 mod p
y 3 = s(x 1 −x 3 )−y 1 mod p, and
s=
[(3x 1 2 + a)/ 2y 1 ] mod p, for P = Q
C. The Discrete Logarithm
[(y 2 - y 1 )/(x 2 -x 1 )] mod p, for P ≠ Q
The security due to ECC relies on the difficulty of Discrete
Logarithm Problem in elliptic fields. If P and Q are two points
on any elliptic curve, so that Q = kP, where k is a scalar, then
it is easy to obtain Q when we know k and P but hard to know
k even if we know P and Q as k should be large. This k is
Results The lab experiment was done in two parts, one with the NAND, NOR, XOR and Hex Inverters and the other with a 7483 full adder gate, both will verify the truth table when two input bits and a carry are added together. The circuits were built by examining the 1 bits through a K-Map to create a Boolean expression for the sum and carry. The Boolean expression for the sum was A⊕B⊕C and the carry as AB+BC_in+AC_in. From these two expressions, we notice that we must use two exclusive-ORs gates in the sum inputs for A, B, and C. For the sum, we have to use NOR and NAND (the only available gates from the lab manual).
(-2 mod 13) (88 mod 6) x + (-10 div 6) y = (3 div 13) ( 24 = 7 * 3 + 3 ) x + ( 25 = 7 * 3 + 4 ) y
1. C1 then performs a permutation on vector [Y] and sends it to C2. C2 decrypts the vector and informs C1 where the distinct bit is located. By performing reverse permutation C1 knows precisely where the bit flip occurs and the two key bits that must be compared 2.
[■(1&1&-2@3&-1&0@3&3&1)] □(→┴(R_2-3R_1&R_3-3R_1 ) ) [■(1&1&-2@0&-4&6@0&0&7)] □(→┴(-1/4 R_2 ) ) [■(1&1&-2@0&1&2⁄3@0&0&7)] →┴(R_1-R_2 ) [■(1&0&(-8)⁄3@0&1&2⁄3@0&0&7)] →┴( 1/7 R_2 )
-x \) And \( \ r(x) = x \) Using the inverse steps introduced in Task 1a the process will be as follows \( \ r(x) = -x \)
For example; M = 4 bits, N = 16 bits If P(j) = 1 (Propagation); then Group(j) will be skipped X(j): m-bits of group (j) Y(j): m-bits of group (j) Cin(j): Carry in to group(j) Cout(j) = Cin(j+1): Carry out of group(j) = Carry in to next group(j+1) (j): Group(j) consisting of m-bits numbers to add Fig 5.14: Carry Skip Adder Block diagram Table 5.3: Carry out Cases Table Case Xi Yi Xi + Yi Ci+1 Comment Ci = 0 Ci = 1 1 0 0 0
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