A power point presentation on the topic “Elliptic Curve Cryptography ( ECC )” with a total of 11 slides.

## Elliptic Curve Cryptography

**Contents**show

### Elliptic Curves in Cryptography

Elliptic Curves (EC) Systems as applied to cryptography were first proposed in 1985 independently by Neal Koblitz and Victor Miller.

The discrete logarithm problem on elliptic curve groups is to believed to be more difficult than the corresponding problem in the (multiplicative groups of nonzero elements of) the underlying finite field.

### What is Elliptic Curve Cryptography (ECC) ?

Elliptic Curve Cryptography (ECC) is a public – key cryptosystem just like RSA, Rabin and El Gamal.

Every user has a public and private key.

Public key is used for encryption/signature verification Private key is used for decryption/signature generation.

Elliptic Curves are used as an extension to other current cryptosystems.

- Elliptic Curve Diffie-Hellman Key Exchange
- Elliptic Curve Digital Signature Algorithm.

### Using Elliptic Curves in Cryptography

The central part of any cryptosystem involving elliptic curves is the elliptic group.

All public key cryptosystems have some underlying mathematical operation.

RSA has exponential (raising the message or ciphertext to the public or private values )

ECC has point multiplication (repeated addition of two points).

### Why use ECC ?

We examine the algorithms used to solve these problems

- How do we analyse cryptosystems ?
- How difficult is the underlying problem that it is based upon RSA, DH, ECC.
- How do we measure Difficulty ?

### Security of ECC

To protect a 128 bit AES key it would take a

- RSA Key Size : 3072 bits
- ECC Key Size : 256 bits

How do we strengthen RSA ? – Increase the key length

Table : NIST guidelines for public key sizes for AES.

ECC KEY Size (Bits) | RSA Key Size (Bits) | Key Size Ratio | AES Key Size ( Bits ) |

163 | 1024 | 1 : 6 | – |

256 | 3072 | 1 : 12 | 128 |

384 | 7680 | 1 : 20 | 192 |

512 | 15360 | 1 : 30 | 256 |

### Applications of ECC

Many devices are small and have limited storage and computational power.

Where can we apply ECC ?

- Wireless Communication Devices
- Smart Cards
- Web Servers that need to handle many encryption sessions
- Any application where security is needed by lacks power, storage and computational power that is necessary for out current cryptosystems

### Benefits of ECC

- Confidentiality
- Integrity
- Authentication
- Non-repudiation
- Shorter Key lengths : Encryption, Decryption and Signature Verification speed up Storage and bandwidth savings