A Like ELGAMAL Cryptosystem But Resistant To Post-Quantum Attacks

The Modulo 1 Factoring Problem (M1FP) is an elegant mathematical problem which could be exploited to design safe cryptographic protocols and encryption schemes that resist to post quantum attacks. The ELGAMAL encryption scheme is a well-known and efficient public key algorithm designed by Taher ELGAMAL from discrete logarithm problem. It is always highly used in Internet security and many other applications after a large number of years. However, the imminent arrival of quantum computing threatens the security of ELGAMAL cryptosystem and impose to cryptologists to prepare a resilient algorithm to quantum computer-based attacks. In this paper we will present a like-ELGAMAL cryptosystem based on the M1FP NP-hard problem. This encryption scheme is very simple but efficient and supposed to be resistant to post quantum attacks

DoubleMod and SingleMod: Simple Randomized Secret-Key Encryption with Bounded Homomorphicity

An encryption relation f Z Z with decryption function f 1 is “group-homomorphic” if, for any suitable plaintexts x1 and x2, x1+x2 = f 1( f (x1)+f (x2)). It is “ring-homomorphic” if furthermore x1x2 = f 1( f (x1) f (x2)); it is “field-homomorphic” if furthermore 1=x1 = f 1( f (1=x1)). Such relations would support oblivious processing of encrypted data. We propose a simple randomized encryption relation f over the integers, called DoubleMod, which is “bounded ring-homomorphic” or what some call ”somewhat homomorphic.” Here, “bounded” means that the number of additions and multiplications that can be performed, while not allowing the encrypted values to go out of range, is limited (any pre-specified bound on the operation-count can be accommodated). Let R be any large integer. For any plaintext x 2 ZR, DoubleMod encrypts x as f (x) = x + au + bv, where a and b are randomly chosen integers in some appropriate interval, while (u; v) is the secret key. Here u > R2 is a large prime and the smallest prime factor of v exceeds u. With knowledge of the key, but not of a and b, the receiver decrypts the ciphertext by computing f 1(y) = (y mod v) mod u. DoubleMod generalizes an independent idea of van Dijk et al. 2010. We present and refine a new CCA1 chosen-ciphertext attack that finds the secret key of both systems (ours and van Dijk et al.’s) in linear time in the bit length of the security parameter. Under a known-plaintext attack, breaking DoubleMod is at most as hard as solving the Approximate GCD (AGCD) problem. The complexity of AGCD is not known. We also introduce the SingleMod field-homomorphic cryptosystems. The simplest SingleMod system based on the integers can be broken trivially. We had hoped, that if SingleMod is implemented inside non-Euclidean quadratic or higher-order fields with large discriminants, where GCD computations appear di cult, it may be feasible to achieve a desired level of security. We show, however, that a variation of our chosen-ciphertext attack works against SingleMod even in non-Euclidean fields

Are Your Credit Cards Safe from Me? Cracking RSA Cryptography

I will begin with a brief overview of the history of cryptography and then specifically look at RSA cryptography. RSA is used for everything from secret communications, to wire transfers between banks, to transmitting your credit card information when you buy items online. I will go over three different factoring methods used to crack RSA cryptography - Pollard\u27s Rho method, Fermat\u27s method, and the continued fractions method. I will discuss the advantages and disadvantages of each, and conclude whether your credit cards are safe from me

Are Your Credit Cards Safe from Me? Cracking RSA Cryptography

I will begin with a brief overview of the history of cryptography and then specifically look at RSA cryptography. RSA is used for everything from secret communications, to wire transfers between banks, to transmitting your credit card information when you buy items online. I will go over three different factoring methods used to crack RSA cryptography - Pollard\u27s Rho method, Fermat\u27s method, and the continued fractions method. I will discuss the advantages and disadvantages of each, and conclude whether your credit cards are safe from me

Homomorphic Encryption and the Approximate GCD Problem

With the advent of cloud computing, everyone from Fortune 500 businesses to personal consumers to the US government is storing massive amounts of sensitive data in service centers that may not be trustworthy. It is of vital importance to leverage the benefits of storing data in the cloud while simultaneously ensuring the privacy of the data. Homomorphic encryption allows one to securely delegate the processing of private data. As such, it has managed to hit the sweet spot of academic interest and industry demand. Though the concept was proposed in the 1970s, no cryptosystem realizing this goal existed until Craig Gentry published his PhD thesis in 2009. In this thesis, we conduct a study of the two main methods for construction of homomorphic encryption schemes along with functional encryption and the hard problems upon which their security is based. These hard problems include the Approximate GCD problem (A-GCD), the Learning With Errors problem (LWE), and various lattice problems. In addition, we discuss many of the proposed and in some cases implemented practical applications of these cryptosystems. Finally, we focus on the Approximate GCD problem (A-GCD). This problem forms the basis for the security of Gentry\u27s original cryptosystem but has not yet been linked to more standard cryptographic primitives. After presenting several algorithms in the literature that attempt to solve the problem, we introduce some new algorithms to attack the problem

The zheng-seberry public key cryptosystem and signcryption

In 1993 Zheng-Seberry presented a public key cryptosystem that was considered efficient and secure in the sense of indistinguishability of encryptions (IND) against an adaptively chosen ciphertext adversary (CCA2). This thesis shows the Zheng-Seberry scheme is not secure as a CCA2 adversary can break the scheme in the sense of IND. In 1998 Cramer-Shoup presented a scheme that was secure against an IND-CCA2 adversary and whose proof relied only on standard assumptions. This thesis modifies this proof and applies it to a modified version of the El-Gamal scheme. This resulted in a provably secure scheme relying on the Random Oracle (RO) model, which is more efficient than the original Cramer-Shoup scheme. Although the RO model assumption is needed for security of this new El-Gamal variant, it only relies on it in a minimal way