Factoring the Modulus of Type N = p2q by Finding Small Solutions of the Equation er − (Ns + t) = αp2 + βq2

The modulus of type N=p2q is often used in many variants of factoring-based cryptosystems due to its ability to fasten the decryption process. Faster decryption is suitable for securing small devices in the Internet of Things (IoT) environment or securing fast-forwarding encryption services used in mobile applications. Taking this into account, the security analysis of such modulus is indeed paramount. This paper presents two cryptanalyses that use new enabling conditions to factor the modulus N=p2q of the factoring-based cryptosystem. The first cryptanalysis considers a single user with a public key pair (e,N) related via an arbitrary relation to equation er−(Ns+t)=αp2+βq2, where r,s,t are unknown parameters. The second cryptanalysis considers two distinct cases in the situation of k-users (i.e., multiple users) for k≥2, given the instances of (Ni,ei) where i=1,…,k. By using the lattice basis reduction algorithm for solving simultaneous Diophantine approximation, the k-instances of (Ni,ei) can be successfully factored in polynomial time.

Q-NTRU Cryptosystem for IoT Applications

Interesting in the Internet of things (IoT) has begun to grow rapidly since it deals with the everyday needs of humans and becomes dealing with a huge amount of personal information. This expansion is accompanied by a number of challenges; one of them is the need for solving the problem of security challenges by using algorithms with high security and the adversaries unable to attack them. But such algorithms need high computation power. On the other hand, the Internet of things has limited resources. Therefore, high security cryptosystem with low computation power is needed. NTRU (Nth-degree TRUncated polynomial ring) is one of lattice-based cryptosystems that meets these requirements. However, this system has weak points, including the ability to attack it under certain condition using Lenstra–Lenstra–Lovász lattice basis reduction algorithm (LLL algorithm) to discover either the original secret key, or an alternative secret key which is useful to decrypt the cipher texts. In this paper, modifications are made on the NTRU cryptosystem algorithm to ensure that the attack by using Lenstra–Lenstra–Lovász algorithm can be thwarted by adding a new parameter with a variable value. The implementation results showed that this modification gives NTRU resistance against this attack.