3.3 Elliptic curve integrated encryption scheme (ECIES)

ECIES combines elliptic curve cryptography with symmetric encryption for secure messaging. It uses a Diffie-Hellman key exchange on elliptic curves to create a shared secret key, which then encrypts and authenticates messages.

The scheme involves key generation, encryption, and decryption steps. It provides strong security based on the Elliptic Curve Discrete Logarithm Problem, making it suitable for various applications requiring high-level protection and efficiency.

Overview of ECIES

• Elliptic Curve Integrated Encryption Scheme (ECIES) is a hybrid encryption scheme that combines the security of elliptic curve cryptography with the efficiency of symmetric encryption
• ECIES provides a standardized way to securely encrypt and decrypt messages using elliptic curve cryptography, ensuring confidentiality, integrity, and authentication
• The scheme leverages the Diffie-Hellman key exchange protocol on elliptic curves to establish a shared secret key, which is then used for symmetric encryption and authentication

Definition of ECIES

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• ECIES is a public-key encryption scheme that utilizes elliptic curve cryptography to encrypt and decrypt messages
• It combines the Elliptic Curve Diffie-Hellman (ECDH) key exchange with symmetric encryption and message authentication
• ECIES provides semantic security, which means that an adversary cannot learn any information about the plaintext from the ciphertext without knowing the private key

Components of ECIES

• ECIES consists of three main components: key generation, encryption, and decryption
• Key generation involves generating a pair of public and private keys based on an elliptic curve
• Encryption combines the recipient's public key with a randomly generated ephemeral key to derive a shared secret, which is then used to encrypt the message and compute an authentication tag
• Decryption uses the recipient's private key and the ephemeral key to derive the shared secret, decrypt the message, and verify the authentication tag

ECIES encryption process

• The ECIES encryption process involves several steps to securely encrypt a message using the recipient's public key and a randomly generated ephemeral key

Key generation in ECIES

• The sender generates an ephemeral key pair $(k, R)$, where $k$ is a random integer and $R = kG$ is a point on the elliptic curve ($G$ is the base point)
• The sender also obtains the recipient's public key $Q$, which is a point on the elliptic curve

Encryption steps in ECIES

• The sender computes a shared secret point $S = kQ$ using the ephemeral private key $k$ and the recipient's public key $Q$
• The shared secret point $S$ is then passed through a key derivation function (KDF) to derive a symmetric encryption key $K_{enc}$ and a MAC key $K_{mac}$
• The sender encrypts the plaintext message $m$ using the symmetric encryption key $K_{enc}$ and a chosen symmetric encryption algorithm (e.g., AES) to obtain the ciphertext $c$
• The sender computes an authentication tag $t$ over the ciphertext $c$ using the MAC key $K_{mac}$ and a chosen MAC algorithm (e.g., HMAC)

Ciphertext structure in ECIES

• The final ciphertext in ECIES consists of three components: the ephemeral public key $R$, the encrypted message $c$, and the authentication tag $t$
• The ciphertext is represented as $(R, c, t)$, where $R$ is a point on the elliptic curve, $c$ is the encrypted message, and $t$ is the authentication tag
• The ciphertext can be transmitted to the recipient, who can use their private key to decrypt the message and verify its authenticity

ECIES decryption process

• The ECIES decryption process involves recovering the plaintext message from the ciphertext using the recipient's private key and the ephemeral public key included in the ciphertext

Decryption steps in ECIES

• The recipient receives the ciphertext $(R, c, t)$, where $R$ is the ephemeral public key, $c$ is the encrypted message, and $t$ is the authentication tag
• The recipient computes the shared secret point $S = dR$ using their private key $d$ and the ephemeral public key $R$
• The shared secret point $S$ is passed through the same key derivation function (KDF) used in the encryption process to derive the symmetric encryption key $K_{enc}$ and the MAC key $K_{mac}$
• The recipient verifies the authenticity of the ciphertext by computing the authentication tag $t'$ over the ciphertext $c$ using the MAC key $K_{mac}$ and comparing it with the received tag $t$. If the tags match, the ciphertext is considered authentic

Plaintext recovery in ECIES

• If the authentication tag is verified successfully, the recipient decrypts the ciphertext $c$ using the symmetric encryption key $K_{enc}$ and the chosen symmetric encryption algorithm (e.g., AES) to obtain the plaintext message $m$
• The recipient has now successfully recovered the original plaintext message $m$ from the ciphertext $(R, c, t)$
• If the authentication tag verification fails, the recipient should discard the ciphertext as it may have been tampered with or corrupted during transmission

Security of ECIES

• ECIES provides strong security guarantees based on the hardness of the Elliptic Curve Discrete Logarithm Problem (ECDLP) and the security of the underlying symmetric encryption and MAC algorithms

Assumptions in ECIES security

• The security of ECIES relies on the assumption that the ECDLP is computationally infeasible to solve
• This means that given a point $Q$ on the elliptic curve, it is extremely difficult to find the scalar $k$ such that $Q = kG$, where $G$ is the base point
• ECIES also assumes that the underlying symmetric encryption algorithm (e.g., AES) and MAC algorithm (e.g., HMAC) are secure and resistant to known attacks

Security proofs for ECIES

• ECIES has been proven to provide indistinguishability under chosen plaintext attack (IND-CPA) and indistinguishability under chosen ciphertext attack (IND-CCA) security
• IND-CPA security ensures that an adversary cannot distinguish between the encryptions of two different messages, even if they can obtain encryptions of chosen plaintexts
• IND-CCA security provides an additional layer of protection, ensuring that an adversary cannot learn any information about the plaintext even if they can obtain decryptions of chosen ciphertexts (except for the challenge ciphertext)
• The security proofs of ECIES rely on the assumptions mentioned earlier and the security of the underlying primitives

Comparison of ECIES vs other schemes

• ECIES offers several advantages over other public-key encryption schemes, such as RSA and ElGamal
• Elliptic curve cryptography provides higher security with smaller key sizes compared to RSA, reducing storage and transmission overhead
• ECIES is more efficient in terms of computation compared to schemes like ElGamal, which requires multiple exponentiations
• ECIES also provides built-in message authentication, eliminating the need for a separate MAC algorithm

Applications of ECIES

• ECIES has found widespread adoption in various domains that require secure communication and data protection

Use cases for ECIES

• ECIES is commonly used in secure messaging applications to encrypt and decrypt messages between users
• It is also employed in secure email communication, ensuring the confidentiality and integrity of email messages
• ECIES is used in blockchain and cryptocurrency systems to secure transactions and protect user privacy
• Internet of Things (IoT) devices and smart contracts often rely on ECIES for secure communication and data exchange

Advantages of ECIES in applications

• ECIES provides strong security guarantees based on the hardness of the ECDLP, making it suitable for applications that require high levels of security
• The small key sizes in elliptic curve cryptography reduce storage and transmission overhead, making ECIES efficient for resource-constrained devices
• The built-in message authentication in ECIES eliminates the need for a separate MAC algorithm, simplifying the implementation and reducing the attack surface
• ECIES is flexible and can be used with various elliptic curves and symmetric encryption algorithms, allowing developers to choose the most suitable parameters for their specific application

Limitations of ECIES in applications

• ECIES requires the secure distribution and management of public keys, which can be challenging in large-scale systems
• The performance of ECIES may be slower compared to symmetric encryption algorithms, especially for large messages
• ECIES relies on the security of the underlying elliptic curve and symmetric encryption algorithms, so any weaknesses discovered in these primitives can impact the security of ECIES
• Implementing ECIES correctly and securely requires careful attention to details, such as proper key generation, secure random number generation, and protection against side-channel attacks

Variants and extensions of ECIES

• Several variants and extensions of ECIES have been proposed to address specific security requirements and improve efficiency

Modified ECIES schemes

• Ephemeral-Static ECIES (ES-ECIES) is a variant that uses a static key pair for the recipient instead of generating a new ephemeral key pair for each encryption, reducing computational overhead
• Elliptic Curve Integrated Encryption Scheme with Key Encapsulation Mechanism (ECIES-KEM) separates the key encapsulation and data encapsulation steps, allowing for more flexibility in key management
• Schnorr ECIES (SECIES) replaces the ECDH key exchange with Schnorr signatures, providing additional authentication and non-repudiation properties

Integration of ECIES with other primitives

• ECIES can be combined with other cryptographic primitives to enhance security or provide additional functionalities
• ECIES can be integrated with digital signatures (e.g., ECDSA) to provide authentication and non-repudiation of encrypted messages
• ECIES can be used in conjunction with key derivation functions (e.g., HKDF) to derive multiple keys for different purposes (encryption, authentication, etc.) from the shared secret
• ECIES can be combined with padding schemes (e.g., OAEP) to provide additional security against chosen-ciphertext attacks

Standardization of ECIES

• ECIES has been standardized by various organizations to ensure interoperability and promote secure implementations
• The ANSI X9.63 standard defines ECIES and provides guidelines for its implementation
• The IEEE 1363a standard includes ECIES as one of the recommended public-key encryption schemes
• The SECG (Standards for Efficient Cryptography Group) has published SEC 1, which specifies ECIES and provides test vectors for implementation validation
• The NIST (National Institute of Standards and Technology) has included ECIES in its recommendations for key management and public-key cryptography