In this tutorial you will learn the elliptic curve cryptography algorithm it’s various encryption algorithms, digital signature algorithm, key agreement algorithms, advantages of Elliptic Curve Cryptography, and a comparison between ECC and RSA algorithm.

- Elliptic Curve Cryptography (ECC) is a public-key cryptographic algorithm based on the algebraic structure of elliptic curves over finite fields.
- It is widely used in securing sensitive data in various applications using encryption and decryption techniques.
- ECC is also used for digital signature to verify the authenticity of a data.

**Encryption algorithm in ECC**

Below are the encryption algorithms used in Elliptic Curve Cryptography (ECC).

### ECIES (Elliptic Curve Integrated Encryption Scheme)

- ECIES is an encryption scheme in ECC that combines symmetric encryption with ECC for secure data transmission.
**ECIES**uses ECC for key agreement to establish a shared secret key which is then used for symmetric encryption of the actual data.- ECIES provides a good balance between security and efficiency, making it suitable for various cryptographic applications.

## EEECC (EC-based ElGamal)

- EEECC algorithm is a variant of the ElGamal encryption scheme that uses ECC for key exchange and encryption.
- It employs ECC operations like point multiplication and addition to generate shared secrets and encrypt data securely.
- EEECC is commonly used in scenarios where ECC-based encryption and key exchange are desired.

## EC ElGamal Encryption

- This is a simplified version of ElGamal encryption specifically tailored for elliptic curves.
- It involves selecting a random scalar
*k*, computing 𝑘×𝐺 to generate an ephemeral key pair, and then using this key pair for encryption. - EC ElGamal encryption provides confidentiality and is often used in ECC-based cryptographic protocols.

## Encryption algorithm in ECC using java.

The following Java source code demonstrates the use of the ECC encryption algorithm for encrypting and decrypting data.

```
package asymmetricCryptography;
import org.bouncycastle.jce.provider.BouncyCastleProvider;
import javax.crypto.Cipher;
import java.security.*;
import java.security.spec.ECGenParameterSpec;
import java.util.Scanner;
public class EccExample {
public static void main(String[] args) throws Exception {
Security.addProvider(new BouncyCastleProvider());
// STEP 1: Ask for user input from keyboard
Scanner in = new Scanner(System.in);
System.out.println("Enter a message to encrypt using ECC:");
String userMessage = in.nextLine();
// STEP 2: Generate an ECC keypair
KeyPair keypair = generateKeyPair();
PrivateKey privateKey = keypair.getPrivate();
PublicKey publicKey = keypair.getPublic();
// Step 3: Encryption on our message
byte [] cipherText = encrypt(userMessage, publicKey);
System.out.println("Original message is: " + userMessage);
System.out.println("Encrypted message using EC is " + new String(cipherText));
// Step 4: Decryption
String originalText = decrypt(cipherText, privateKey);
System.out.println("The decoded information using private key is " + originalText);
}
public static KeyPair generateKeyPair() throws Exception {
KeyPairGenerator keyPairGenerator = KeyPairGenerator.getInstance("EC", "BC");
ECGenParameterSpec ecSpec = new ECGenParameterSpec("secp256k1");
keyPairGenerator.initialize(ecSpec);
return keyPairGenerator.generateKeyPair();
}
public static byte[] encrypt(String message, PublicKey publicKey) throws Exception {
Cipher cipher = Cipher.getInstance("ECIES", "BC");
cipher.init(Cipher.ENCRYPT_MODE, publicKey);
return cipher.doFinal(message.getBytes());
}
public static String decrypt(byte [] cipherText, PrivateKey privateKey) throws Exception {
Cipher cipher = Cipher.getInstance("ECIES", "BC");
cipher.init(Cipher.DECRYPT_MODE, privateKey);
return new String(cipher.doFinal(cipherText));
}
}
```

**Digital signature algorithm in ECC**

Digital signatures are employed to verify the authenticity, integrity, and non-repudiation of information. Two widely used digital signature algorithms in elliptic curve cryptography include ECDSA and EdDSA algorithm.

- Elliptic Curve Digital Signature Algorithm (ECDSA)
- Edwards-curve Digital Signature Algorithm (EdDSA).

### Elliptic Curve Digital Signature Algorithm (ECDSA)

ECDSA is a sophisticated public-key cryptography encryption algorithm. It leverages the algebraic properties of elliptic curves within finite fields.

### Edwards-curve Digital Signature Algorithm (EdDSA)

- EdDSA serves as an alternative to ECDSA, designed to provide fast public-key digital signatures with improved performance and simplified implementations. Its use of Edwards curves contributes to its efficiency and security, particularly in embedded devices where resource constraints are a concern.
- EdDSA’s design reduces the vulnerability to certain side-channel attacks, enhancing its overall security posture.

### ECC digital signature implementation in java

This Java code below demonstrates the process of generating an ECDSA digital signature on a user defined message using private key and verifying the signature’s authenticity using public key.

```
import java.security.*;
import java.security.spec.ECGenParameterSpec;
import java.security.interfaces.ECPrivateKey;
import java.security.interfaces.ECPublicKey;
import java.util.Base64;
public class ECDSASignatureExample {
public static void main(String[] args) throws Exception {
// Generate ECDSA key pair
KeyPairGenerator keyGen = KeyPairGenerator.getInstance("EC");
// Defin Elliptic curve parameters
ECGenParameterSpec ecSpec = new ECGenParameterSpec("secp256r1");
keyGen.initialize(ecSpec);
KeyPair keyPair = keyGen.generateKeyPair();
PrivateKey privateKey = keyPair.getPrivate();
PublicKey publicKey = keyPair.getPublic();
// Create a digital signature for an input message annd sign
Signature signature = Signature.getInstance("SHA256withECDSA");
signature.initSign(privateKey);
String data = "Hello, world!";
signature.update(data.getBytes());
byte[] digitalSignature = signature.sign();
// Verify the digital signature using public key
signature.initVerify(publicKey);
signature.update(data.getBytes());
boolean isVerified = signature.verify(digitalSignature);
// Print results
System.out.println("Original data: " + data);
System.out.println("Digital Signature: " + Base64.getEncoder().encodeToString(digitalSignature));
System.out.println("Is Signature Verified: " + isVerified);
}
}
```

**Key Agreement algorithm in ECC**

### Elliptic-curve Diffie-Hellman (ECDH)

- ECDH is a key agreement protocol used to establish a shared secret between two parties over an insecure communication channel.
- Each party possesses an elliptic-curve public-private key pair.
- The protocol involves the following steps:
- Both parties exchange their public keys.
- They each compute a shared secret using their private key and the received public key.
- This shared secret can be used directly as a key or used to derive another key for encryption purposes.
- ECDH provides forward secrecy and is widely used in secure communication protocols.

### Fully Hashed Menezes-Qu-Vanstone (FHMQV)

- FHMQV is an authenticated key agreement protocol based on the Diffie-Hellman scheme.
- It is an extension of the MQV scheme and offers protection against active attackers.
- The protocol can be adapted to work in any finite group, including elliptic curve groups, where it is known as elliptic curve MQV (ECMQV).
- ECMQV enhances security by using additional hashing and authentication mechanisms.
- It enables parties to establish a shared secret securely, ensuring the confidentiality and integrity of communications.

**Elliptic curve cryptography algorithm implementation in java**

**Advantage of Elliptic Curve Cryptography (ECC)**

Elliptic Curve Cryptography (ECC) offers numerous benefits over traditional cryptographic systems, making it a popular choice for protecting sensitive data and communications across different applications. The advantages of ECC include:

### Strong Security with Smaller Key Sizes

- Elliptic Curve Cryptography (ECC) provides an equivalent or even stronger security compared to other cryptographic algorithms like RSA but with smaller key sizes.
- The advantage of using smaller key size is crucial for resource-constrained environments like mobile devices and IoT devices, where storage and computational power are limited.

### Resistance to Quantum Attacks

- Elliptic Curve Cryptography (ECC) is inherently resistant to quantum attacks due to the complexity of solving the Elliptic Curve Discrete Logarithm Problem (ECDLP) on elliptic curves.
- Quantum computers can break regular cryptographic systems, but when it comes to ECC, they struggle with the math, which makes ECC more secure for the long run.

### Efficient Resource Usage

- Elliptic Curve Cryptography (ECC) requires a less computational power and bandwidth compared to other algorithms which lead to a faster cryptographic operations and reduced resource consumption.

### Versatility and Flexibility

- Elliptic Curve Cryptography can be applied to various cryptographic operations like encryption, authentication, digital signatures, and key exchange protocols.

### Lower Operational Costs

- The smaller key sizes and reduced computational overhead of ECC result in lower operational costs for implementing and managing cryptographic systems.

### Standardization and Interoperability

- ECC has been standardized by organizations such as the National Institute of Standards and Technology (NIST), ensuring interoperability and compatibility across different systems and platforms.

### Scalability and Future Proofing

- Elliptic Curve Cryptography is scalable, allowing for the adaptation of key sizes and security parameters to meet evolving security requirements and technological advancements.
- ECC resilience to quantum attacks and efficient resource usage make ECC a future-proof cryptographic solution for securing digital assets and communications.

** Comparison between ECC vs RSA **

Aspect | ECC | RSA |
---|---|---|

Key Size | Smaller key size, 256 bits for equivalent security of RSA | Larger key size 3072 bits for equivalent security to ECC |

Computational Efficiency | Requires less computational power and bandwidth.. | Requires more computational power. |

Security Strength | ECC is strong against various attacks including quantum attacks. | RSA security relies on the difficulty of factoring large prime numbers.. |

Resource Consumption | Consumes less memory and bandwidth | Consumes more resources and may not be ideal in resource limited environment. |

Key Management | Employes simplified key management and storage techniques. | More complex key management processes in RSA. |

The above table summarizes the key differences between Elliptic Curve Cryptography (ECC) and Rivest-Shamir-Adleman (RSA) in terms of* key size, computational efficiency, security strength, resource consumption, and key management.*