Objectives of modern cryptography include:.

Protecting information is vital to our way of life. Coding theory along with modern cryptography is crucial in achieving this and offering valuable protecting against things such as fraud and identity theft. Already a member? Coding Theory: The wide world of cryptography. When learning about coding theory, remember these three associated terms: Data Compression: the concept of data compression is about the most efficient way of encoding information so it takes up as little space as possible and this can be accomplished via removing redundancy from the data via source encoding.

Error Correcting Codes: Error correcting codes are used to improve communication reliability over noisy channels which is accomplished by adding redundancy. Cryptography sometimes called cryptology : Cryptography concerns the security, privacy, and confidentiality of information transmitted over a secure channel. The CIA tirad plays an important role in cryptography.

Objectives of modern cryptography include: Confidentiality Data integrity Authentication Non-repudiation Secret sharing Protecting information is vital to our way of life. Cryptography, on the other hand, protects communication over insecure channels.

Modern cryptography relies heavily on mathematics, computer science and most of all cleverness. In a good cryptographic system , if one bit is changed in the ciphertext, enough bits are modified in the corresponding plaintext, and this makes it unreadable. Therefore, it is important to have ways of detecting and correcting errors that might happen in case a ciphertext is transmitted. For example, CD players, fax machines, computer hard drives and other devices which process digital data.

Error correction codes are usually used to solve this problem. Although coding theory — communication over noisy channels is technically not a part of cryptography — communication over insecure channels, error correcting codes can be used to construct a public key system.

## Coding and cryptography

All communication channels have some degree of noise, also known as interference. It is mostly caused by various things like adjacent channels, deterioration of equipment and electric impulses amongst others. The noise can interfere with data transmission at times. Just like having a conversation in a noisy room is hard, so does data transmission if the channel becomes noisier.

For one to have a conversation in a noisy room, they must shout or repeat themselves. For the same to take place over a noisy channel, we are required to add some redundancy to the transmission, for the recipient to be able to reconstruct the message. The following example shows how redundancy techniques can be used. Here, the original message is replaced by code words which have redundancy built on them.

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For example, how many pennies can be packed into a circle on a tabletop, or in 3 dimensions, how many marbles can be packed into a globe. Other considerations enter the choice of a code. For example, hexagon packing into the constraint of a rectangular box will leave empty space at the corners. As the dimensions get larger, the percentage of empty space grows smaller. But at certain dimensions, the packing uses all the space and these codes are the so-called "perfect" codes. The only nontrivial and useful perfect codes are the distance-3 Hamming codes with parameters satisfying 2 r — 1, 2 r — 1 — r , 3 , and the [23,12,7] binary and [11,6,5] ternary Golay codes.

Another code property is the number of neighbors that a single codeword may have. First we pack the pennies in a rectangular grid. Each penny will have 4 near neighbors and 4 at the corners which are farther away. In a hexagon, each penny will have 6 near neighbors. When we increase the dimensions, the number of near neighbors increases very rapidly.

The result is the number of ways for noise to make the receiver choose a neighbor hence an error grows as well. This is a fundamental limitation of block codes, and indeed all codes. It may be harder to cause an error to a single neighbor, but the number of neighbors can be large enough so the total error probability actually suffers. Properties of linear block codes are used in many applications.

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For example, the syndrome-coset uniqueness property of linear block codes is used in trellis shaping, [7] one of the best known shaping codes. This same property is used in sensor networks for distributed source coding, and in lossy compression of noisy sparse sources [8]. The idea behind a convolutional code is to make every codeword symbol be the weighted sum of the various input message symbols.

This is like convolution used in LTI systems to find the output of a system, when you know the input and impulse response. So we generally find the output of the system convolutional encoder, which is the convolution of the input bit, against the states of the convolution encoder, registers. Fundamentally, convolutional codes do not offer more protection against noise than an equivalent block code.

In many cases, they generally offer greater simplicity of implementation over a block code of equal power. The encoder is usually a simple circuit which has state memory and some feedback logic, normally XOR gates. The decoder can be implemented in software or firmware. The Viterbi algorithm is the optimum algorithm used to decode convolutional codes. There are simplifications to reduce the computational load. They rely on searching only the most likely paths. Although not optimum, they have generally been found to give good results in low noise environments.

Convolutional codes are used in voiceband modems V. Cryptography or cryptographic coding is the practice and study of techniques for secure communication in the presence of third parties called adversaries. Modern cryptography exists at the intersection of the disciplines of mathematics , computer science , and electrical engineering. Applications of cryptography include ATM cards , computer passwords , and electronic commerce. Cryptography prior to the modern age was effectively synonymous with encryption , the conversion of information from a readable state to apparent nonsense.

### Some Applications of Coding Theory in Cryptography (1995)

The originator of an encrypted message shared the decoding technique needed to recover the original information only with intended recipients, thereby precluding unwanted persons from doing the same. Modern cryptography is heavily based on mathematical theory and computer science practice; cryptographic algorithms are designed around computational hardness assumptions , making such algorithms hard to break in practice by any adversary.

It is theoretically possible to break such a system, but it is infeasible to do so by any known practical means. These schemes are therefore termed computationally secure; theoretical advances, e.

There exist information-theoretically secure schemes that provably cannot be broken even with unlimited computing power—an example is the one-time pad —but these schemes are more difficult to implement than the best theoretically breakable but computationally secure mechanisms. A line code also called digital baseband modulation or digital baseband transmission method is a code chosen for use within a communications system for baseband transmission purposes. Line coding is often used for digital data transport. Line coding consists of representing the digital signal to be transported by an amplitude- and time-discrete signal that is optimally tuned for the specific properties of the physical channel and of the receiving equipment.

## Introduction to Cryptography with Coding Theory Solutions

The waveform pattern of voltage or current used to represent the 1s and 0s of a digital data on a transmission link is called line encoding. The common types of line encoding are unipolar , polar , bipolar , and Manchester encoding. Another concern of coding theory is designing codes that help synchronization. A code may be designed so that a phase shift can be easily detected and corrected and that multiple signals can be sent on the same channel.

Another application of codes, used in some mobile phone systems, is code-division multiple access CDMA.

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Each phone is assigned a code sequence that is approximately uncorrelated with the codes of other phones. At the receiver, a demodulation process is performed to recover the data. The properties of this class of codes allow many users with different codes to use the same radio channel at the same time. To the receiver, the signals of other users will appear to the demodulator only as a low-level noise.

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Another general class of codes are the automatic repeat-request ARQ codes. In these codes the sender adds redundancy to each message for error checking, usually by adding check bits. If the check bits are not consistent with the rest of the message when it arrives, the receiver will ask the sender to retransmit the message. All but the simplest wide area network protocols use ARQ.

There is an extensive field of research on this topic because of the problem of matching a rejected packet against a new packet. Is it a new one or is it a retransmission?