Elliptic Curve Integrated Encryption Scheme

The integrated encryption scheme (ECIES) is a hybrid encryption method , the elliptic curves are based. As a hybrid process , it combines an asymmetrical process , which is used to send a symmetrical key, with a symmetrical encryption process , which encrypts the message with this symmetrical key. In the Random Oracle model, ECIES is secure against chosen ciphertext attacks .

Establishing the scheme

The following resources are required:

• KDF ( Key Derivation Function ): a cryptographic hash function that can generate keys of any length
• MAC ( Message Authentication Code )
• A symmetrical encryption method with an encryption algorithm and a decryption algorithm ${\displaystyle E}$ ${\displaystyle D}$

System parameters

• ${\displaystyle \mathbb {F} _{p}}$, Prime number ${\displaystyle p}$
• Elliptic curve E: above the body ${\displaystyle Y^{2}=X^{3}+aX+b}$ ${\displaystyle \mathbb {F} _{p}}$
• ${\displaystyle P\in E(\mathbb {F} _{p})}$with prim ${\displaystyle ord(P)=n}$
• ${\displaystyle h={\frac {\mid E(\mathbb {F} _{p})\mid }{n}}}$

Key generation

A participant chooses a secret key at random and calculates his public key from it . ${\displaystyle A}$ ${\displaystyle d_{A}\in \{1,...,n-1\}}$ ${\displaystyle EK_{A}=d_{A}P}$

Encryption

To encrypt a message with a public key , a Diffie-Hellman key exchange is combined in an elliptical curve with a symmetrical method. ${\displaystyle m\in \{0,1\}^{*}}$ ${\displaystyle EK_{A}}$

1. Pick a random number ${\displaystyle k\in \{1,...,n-1\}}$
2. Calculate and ${\displaystyle R=kP}$ ${\displaystyle Z=hk\cdot EK_{A}}$
3. Find the symmetric keys . is the x-coordinate of ${\displaystyle k_{1}||k_{2}=KDF(Z_{x})}$ ${\displaystyle Z_{x}}$ ${\displaystyle Z}$
4. Calculate and ${\displaystyle C=E_{k_{1}}(m)}$ ${\displaystyle T=MAC_{k_{2}}(C)}$
5. send ${\displaystyle (R,C,T)}$

Decryption

In order to decrypt a cipher with a secret key , the following steps are carried out. ${\displaystyle (R,C,T)}$ ${\displaystyle d_{A}}$

1. Calculate ${\displaystyle Z=hd_{A}R}$
2. Determine the two keys ${\displaystyle k_{1}||k_{2}=KDF(Z_{x})}$
3. Check if is ${\displaystyle T=MAC_{k_{2}}(C)}$
4. Get ${\displaystyle m=D_{k_{1}}(C)}$

Conclusion

ECIES works correctly if it is calculated correctly. There is, this is validated. ${\displaystyle Z}$ ${\displaystyle hd_{A}R=hk\cdot EK_{A}=hd_{A}kP}$

swell

• Alfred J. Menezes, Paul C. van Oorschot and Scott A. Vanstone: Handbook of Applied Cryptography . CRC Press, ISBN 0-8493-8523-7 ( uwaterloo.ca ).
• Victor Shoup: A proposal for an ISO standard for public key encryption , Version 2.1, December 20, 2001 (PDF, 384 kB).