Gives Detail notes about ECC

1. ELLIPTIC CURVE
CRYPTOGRAPHY

2. ELLIPTIC CURVE
An Elliptic curve is a set of
points that satisfy an equation
of the form.
By
Ms.R.Pooja.,B.Sc.,B.Ed
K.R.College of Arts & Science,
Kovilpatti.

3. An elliptic curve is a set of
points that satisfy an equation
of the form

4. Where a and b are
constants and the
Curve is defined over
a finite field
Symmetric X- axis
3 points at max can
be generated.

5. Elliptic curve cryptography (ECC) was
independently discovered and
proposed in 1985 by Neal Koblitz and
Victor Miller. They are credited with
recognizing the potential of using
elliptic curves over finite fields in
cryptographic systems.

6. It is a type of Public – Key cryptography based
on the mathematics of elliptic curves.
It provides high security with smaller key size.
Similar to RSA – makes use of public key
cryptography
ECC algorithm used to perform critical security
functions, including encryption, authentication,
and digital signatures.
ELLIPTIC CURVE
CRYPTOGRAPHY

7. To form a cryptographic system
using elliptic curves, we need to Find
a “hard problem” corresponding to
factoring the product of two primes
Or taking the discrete logarithm.

8. ECC is based on the elliptic curve theory, which generates
keys through the properties of the elliptic curve equation,
compared to the traditional method of factoring very large
prime numbers.

9. X Y
Easy
Hard
TRAPDOOR FUNCTION
Easy if given a trapdoor value ‘t’
Q=KP - Easy
P inverse Q=K -Hard
K=?

10. Example
Problem for ellptic curve
To solve the elliptic curve discrete logarithm problem (ECDLP)

17. P = (16, 5)
2P = (20, 20)
3P = (14, 14)
4P = (19, 20)
5P = (13, 10)
6P = (7, 3)
7P = (8, 7)
8P = (12, 17)
9P = (4, 5) = Q

19. Key exchange using elliptic curve cryptography

20. Elliptic curve Encryption /Decryption
1.First task in this system is to encode the plaintext message m to be sent as
an x-y point pm.It is the point Pm that will be encrypted as a cipher text and
subsequently decrypted.
Note that we cannot simply encode the message as the x or y coordinate of a point,
because not all Such coordinates are in Eq(a,b).
2.

22. Elliptic curve cryptography has several practical applications in the real world.
One example is online banking and payments. When you make an online
purchase with your debit or credit card, your information is often encrypted
using ECC before it’s sent over the internet. This ensures that your information
remains confidential and secure throughout the transaction process.
Another application of ECC is in email encryption. Pretty Good Privacy (PGP) is a
popular email encryption software that can leverage ECC to protect your emails
from being read by anyone other than the intended recipient.
PGP works by generating a public/private key pair for each user. The public key
can be shared with anyone, but the private key must be kept confidential at all
times. To encrypt an email, you simply need the recipient’s public key;
conversely, you’ll need your private key to decrypt an email you’ve received.
Real-world applications of Elliptic Curve Cryptography
ECC is a powerful tool that can protect data, authenticate connections,
and verify integrity in various applications. As more and more of our lives
move online, cryptography is essential to keep our data safe and secure.

24. ADVANTAGES
Shorter key size then RSA.
Higher Security.
Low computational power- Suitable
for smart phone & tablets.