الفهرس 
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Abstract
Elliptic Curve Cryptography (ECC) is one of the prominent public key
methodologies because of its small key size and high security compared to other
methods. Secure applications on limited environments like smart cards present
implementation challenges due to the platform’s limited memory, bandwidth and
computational capacity. The characteristics of ECC techniques make them
suitable to overcome these challenges. This thesis investigates and improves
different aspects of elliptic curve cryptography. It proposes using elliptic curves
over Gaussian integers to speed up the cryptographic computations. It also
introduces an elliptic curve based signcryption algorithm that provides encrypted
message authentication with forward secrecy. The algorithm enables firewalls to
authenticate encrypted messages without having to decrypt them. Next, the thesis
moves to introducing kleptographic attacks on elliptic curves. An algorithm is
presented to add a secretly embedded trapdoor with universal protection into the
elliptic curve discrete logarithm problem and extends this to an attack on elliptic
curve key exchange. The algorithm enables the attacker to covertly extract the
shared key, which implies allowing the attacker to forge signatures and to decrypt
all the communications that involve this key. The research extends this attack to
elliptic curve encryption to enable the attacker to extract the secret messages
without knowing the secret key. The attack is also found possible on elliptic curve
signatures where it can leak the key just to the attacker. Finally an algorithm is
proposed to use the Elliptic Curve Digital Signature Algorithm (ECDSA) for
encrypting subliminal messages without having to use an encryption mechanism.