An Analysis of Modern Cryptosystems

Since the ancient Egyptian empire, man has searched for ways to protect information from getting into the wrong hands. Julius Caesar used a simple substitution cipher to protect secrets. During World War II, the Allies and the Axis had codes that they used to protect information. Now that we have computers at our disposal, the methods used to protect data in the past are ineffective. More recently, computer scientists and mathematicians have been working diligently to develop cryptosystems which will provide absolute security in a computing environment. The three major cryptosystems in use today are DES, RSA, and the Knapsack Cryptosystem. These cryptosystems have been reviewed and the positive and negative aspects of each is discussed. A newcomer to the field of cryptology is the Random Spline Cryptosystem which is discussed in detail

The Interpolating Random Spline Cryptosystem and the Chaotic-Map Public-Key Cryptosystem

The feasibility of implementing the interpolating cubic spline function as encryption and decryption transformations is presented. The encryption method can be viewed as computing a transposed polynomial. The main characteristic of the spline cryptosystem is that the domain and range of encryption are defined over real numbers, instead of the traditional integer numbers. Moreover, the spline cryptosystem can be implemented in terms of inexpensive multiplications and additions. Using spline functions, a series of discontiguous spline segments can execute the modular arithmetic of the RSA system. The similarity of the RSA and spline functions within the integer domain is demonstrated. Furthermore, we observe that such a reformulation of RSA cryptosystem can be characterized as polynomials with random offsets between ciphertext values and plaintext values. This contrasts with the spline cryptosystems, so that a random spline system has been developed. The random spline cryptosystem is an advanced structure of spline cryptosystem. Its mathematical indeterminacy on computing keys with interpolants no more than 4 and numerical sensitivity to the random offset t( increases its utility. This article also presents a chaotic public-key cryptosystem employing a one-dimensional difference equation as well as a quadratic difference equation. This system makes use of the El Gamal’s scheme to accomplish the encryption process. We note that breaking this system requires the identical work factor that is needed in solving discrete logarithm with the same size of moduli

Criptosistemas de clave pública basados en el problema de las mochilas

Un criptosistema de clave pública es un sistema de transmisión de mensajes entre un emisor y un receptor a través de una función de una vía, es decir, una función cuya inversa es muy difícil de calcular sin una información complementaria de la que sólo dispone el receptor legítimo. Una de estas funciones es el problema de las mochilas, que consiste en, dado un conjunto de pesos A y un número grande S, encontrar, si existe, un subconjunto de A tal que la suma de sus elementos sea S. Tanto los pesos como S son números naturales. El trabajo consistirá en una explicación y ejemplos del criptosistema de Merkle-Hellmann (1978), el posterior de Shamir (1982) y un resumen de las variantes surgidas hasta la fecha, dado que el sistema de Merkle-Hellmann ya no es útil.Grado en Matemática

Quadratic compact knapsack public-key cryptosystem

AbstractKnapsack-type cryptosystems were among the first public-key cryptographic schemes to be invented. Their NP-completeness nature and the high speed in encryption/decryption made them very attractive. However, these cryptosystems were shown to be vulnerable to the low-density subset-sum attacks or some key-recovery attacks. In this paper, additive knapsack-type public-key cryptography is reconsidered. We propose a knapsack-type public-key cryptosystem by introducing an easy quadratic compact knapsack problem. The system uses the Chinese remainder theorem to disguise the easy knapsack sequence. The encryption function of the system is nonlinear about the message vector. Under the relinearization attack model, the system enjoys a high density. We show that the knapsack cryptosystem is secure against the low-density subset-sum attacks by observing that the underlying compact knapsack problem has exponentially many solutions. It is shown that the proposed cryptosystem is also secure against some brute-force attacks and some known key-recovery attacks including the simultaneous Diophantine approximation attack and the orthogonal lattice attack

Self Masking for Hardering Inversions

The question whether one way functions (i.e., functions that are easy to compute but hard to invert) exist is arguably one of the central problems in complexity theory, both from theoretical and practical aspects. While proving that such functions exist could be hard, there were quite a few attempts to provide functions which are one way in practice , namely, they are easy to compute, but there are no known polynomial time algorithms that compute their (generalized) inverse (or that computing their inverse is as hard as notoriously difficult tasks, like factoring very large integers). In this paper we study a different approach. We provide a simple heuristic, called self masking, which converts a given polynomial time computable function $f$ into a self masked version $[{f}]$, which satisfies the following: for a random input $x$, $[{f}]^{-1}([{f}](x))=f^{-1}(f(x))$ w.h.p., but a part of $f(x)$, which is essential for computing $f^{-1}(f(x))$ is masked in $[{f}](x)$. Intuitively, this masking makes it hard to convert an efficient algorithm which computes $f^{-1}$ to an efficient algorithm which computes $[{f}]^{-1}$, since the masked parts are available to $f$ but not to $[{f}]$. We apply this technique on variants of the subset sum problem which were studied in the context of one way functions, and obtain functions which, to the best of our knowledge, cannot be inverted in polynomial time by published techniques

Public key cryptosystems : theory, application and implementation

The determination of an individual's right to privacy is mainly a nontechnical matter, but the pragmatics of providing it is the central concern of the cryptographer. This thesis has sought answers to some of the outstanding issues in cryptography. In particular, some of the theoretical, application and implementation problems associated with a Public Key Cryptosystem (PKC).The Trapdoor Knapsack (TK) PKC is capable of fast throughput, but suffers from serious disadvantages. In chapter two a more general approach to the TK-PKC is described, showing how the public key size can be significantly reduced. To overcome the security limitations a new trapdoor was described in chapter three. It is based on transformations between the radix and residue number systems.Chapter four considers how cryptography can best be applied to multi-addressed packets of information. We show how security or communication network structure can be used to advantage, then proposing a new broadcast cryptosystem, which is more generally applicable.Copyright is traditionally used to protect the publisher from the pirate. Chapter five shows how to protect information when in easily copyable digital format.Chapter six describes the potential and pitfalls of VLSI, followed in chapter seven by a model for comparing the cost and performance of VLSI architectures. Chapter eight deals with novel architectures for all the basic arithmetic operations. These architectures provide a basic vocabulary of low complexity VLSI arithmetic structures for a wide range of applications.The design of a VLSI device, the Advanced Cipher Processor (ACP), to implement the RSA algorithm is described in chapter nine. It's heart is the modular exponential unit, which is a synthesis of the architectures in chapter eight. The ACP is capable of a throughput of 50 000 bits per second