Problem statement:A cryptosystem is a way for a sender and a receiver to communicate digitally by which the sender can send receiver any confidential or private message by first encrypting it using the receiver's public key. Upon receiving the encrypted message, the receiver can confirm the originality of the message's contents using his own secret key. Up to now, most of the existing cryptosystems were developed based on a single cryptographic assumption like factoring, discrete logarithms, quadratic residue or elliptic curve discrete logarithm. Although these schemes remain secure today, one day in a near future they may be broken if one finds a polynomial algorithm that can efficiently solve the underlying cryptographic assumption. Approach: By this motivation, we designed a new cryptosystem based on two cryptographic assumptions; quadratic residue and discrete logarithms. We integrated these two assumptions in our encrypting and decrypting equations so that the former depends on one public key whereas the latter depends on one corresponding secret key and two secret numbers. Each of public and secret keys in our scheme determines the assumptions we use. Results: The newly developed cryptosystem is shown secure against the three common considering algebraic attacks using a heuristic security technique. The efficiency performance of our scheme requires 2T exp +2T mul +T hash time complexity for encryption and T exp +2T mul +T srt time complexity for decryption and this magnitude of complexity is considered minimal for multiple cryptographic assumptions-like cryptosystems. Conclusion: The new cryptosystem based on multiple cryptographic assumptions offers a greater security level than that schemes based on a single cryptographic assumption. The adversary has to solve the two assumptions simultaneously to recover the original message from the received corresponding encrypted message but this is very unlikely to happen.

Problem statement:A cryptosystem allows a sender to send any confidential or private message using a receiver's public key and later the receiver confirms the integrity of the received message using his secret key. Currently the existing cryptosystems were developed based on a single hard problem like factoring, discrete logarithm, residuosity, knapsack or elliptic curve discrete logarithm. Although these schemes appear secure, one day in a near future they may be broken if one finds a solution of a single hard problem. Approach: To solve this problem, we developed a new cryptosystem based on two hard problems; factoring and discrete logarithm. We integrated the two problems in our encrypting and decrypting equations so that the former depends on two public keys whereas the latter depends on two corresponding secret keys. Results: The new cryptosystem is shown secure against the most three considering attacks. The efficiency performance of our scheme only requires 3T exp +T mul + T hash time complexity for encryption and 2T exp + T mul time complexity for decryption and this magnitude of complexity is considered minimal for multiple hard problems-like cryptosystems. Conclusion: The new cryptosystem based on multiple hard problems provides longer and higher security level than that schemes based on a single hard problem. The adversary has to solve the two problems simultaneously in order to recover a corresponding plaintext (message) from the received ciphertext (encrypted message).

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Problem statement:The security of elliptic curve cryptosystems are based on elliptic curve discrete logarithm problem (ECDLP). However, if an attacker finds a solution to ECDLP, the elliptic curve-based systems will no longer be secure. Approach: To improve this, we develop a new elliptic curve cryptosystem using one of the old/novel problem in computational number theory; factoring problem (FAC). Specifically, our encrypting and decrypting equations will heavily depends on two public keys and two secret keys respectively. Results: We show that, the newly designed cryptosystem is heuristically secure against various algebraic attacks. The complexity of the scheme shows that the time complexity for each encryption and decryption are given by 299T mul and 270T mul . Conclusion:The new system provides greater security than that system based on a single hard problem. The attacker has not enough resources to solve the two hard problems simultaneously in a polynomial time.

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