# Encryption and Decryption in Digital Communications

• Print
This chapter is from the book

## 17.4 Stream Encryption

Earlier, we defined a one-time pad as an encryption system with a random key, used one time only, that exhibits unconditional security. One can conceptualize a stream encryption implementation of a one-time pad using a truly random key stream (the key sequence never repeats). Thus, perfect secrecy can be achieved for an infinite number of messages, since each message would be encrypted with a different portion of the random key stream. The development of stream encryption schemes represents an attempt to emulate the one-time pad. Great emphasis was placed on generating key streams that appeared to be random, yet could easily be implemented for decryption, because they could be generated by algorithms. Such stream encryption techniques use pseudorandom (PN) sequences, which derive their name from the fact that they appear random to the casual observer; binary

pseudorandom sequences have statistical properties similar to the random flipping of a fair coin. However, the sequences, of course, are deterministic (see Section12.2). These techniques are popular because the encryption and decryption algorithms are readily implemented with feedback shift registers. At first glance it may appear that a PN key stream can provide the same security as the one-time pad, since the period of the sequence generated by a maximum-length linear shift register is 2n - 1 bits, where n is the number of stages in the register. If the PN sequence were implemented with a 50-stage register and a 1-MHz clock rate, the sequence would repeat every 250 - 1 microseconds, or every 35 years. In this era of large-scale integrated (LSI) circuits, it is just as easy to provide an implementation with 100 stages, in which case the sequence would repeat every 4 × 1016 years. Therefore, one might suppose that since the PN sequence does not repeat itself for such a long time, it would appear truly random and yield perfect secrecy. There is one important difference between the PN sequence and a truly random sequence used by a one-time pad. The PN sequence is generated by an algorithm; thus, knowing the algorithm, one knows the entire sequence. In Section 17.4.2 we will see that an encryption scheme that uses a linear feedback shift register in this way is very vulnerable to a known plaintext attack.

### 17.4.1 Example of Key Generation Using a Linear Feedback Shift Register

Stream encryption techniques generally employ shift registers for generating their PN key sequence. A shift register can be converted into a pseudorandom sequence generator by including a feedback loop that computes a new term for the first stage based on the previous n terms. The register is said to be linear if the numerical operation in the feedback path is linear. The PN generator example from Section 12.2 is repeated in Figure 17.13. For this example, it is convenient to number the stages as shown in Figure 17.13, where n = 4 and the outputs from stages 1 and 2 are modulo-2 added (linear operation) and fed back to stage 4. If the initial state of stages (x4, x3, x2, x1) is 1 0 0 0, the succession of states triggered by clock pulses would be 1 0 0 0, 0 1 0 0, 0 0 1 0, 1 0 0 1, 1 1 0 0, and so on. The output sequence is made up of the bits shifted out from the rightmost stage of the register, that is, 1 1 1 1 0 1 0 1 1 0 0 1 0 0 0, where the rightmost bit in this sequence is the earliest output and the leftmost bit is the most recent output. Given any linear feedback shift register of degree n, the output sequence is ultimately periodic.

### 17.4.2 Vulnerabilities of Linear Feedback Shift Registers

An encryption scheme that uses a linear feedback shift register (LFSR) to generate the key stream is very vulnerable to attack. A cryptanalyst needs only 2n bits of plaintext and its corresponding ciphertext to determine the feedback taps, the initial state of the register, and the entire sequence of the code. In general, 2n is very small compared with the period 2n − 1. Let us illustrate this vulnerability with the LFSR example illustrated in Figure 17.13. Imagine that a cryptanalyst who knows nothing about the internal connections of the LFSR manages to obtain 2n = 8 bits of ciphertext and its plaintext equivalent:

Plaintext: 0 1 0 1 0 1 0 1

Ciphertext: 0 0 0 0 1 1 0 0

where the rightmost bit is the earliest received and the leftmost bit is the most recent that was received.

The cryptanalyst adds the two sequences together, modulo-2, to obtain the segment of the key stream, 0 1 0 1 1 0 0 1, illustrated in Figure 17.14. The key stream sequence shows the contents of the LFSR stages at various times. The rightmost border surrounding four of the key bits shows the contents of the shift register at time t1. As we successively slide the “moving” border one digit to the left, we see the shift register contents at times t2, t3, t4,.... From the linear structure of the four-stage shift register, we can write

where x5 is the digit fed back to the input and gi (= 1 or 0) defines the ith feedback connection. For this example, we can thus write the following four equations with four unknowns, by examining the contents of the shift register at the four times shown in Figure 17.14:

The solution of Equations (17.28) is g1 = 1, g2 = 1, g3 = 0, g4 = 0, corresponding to the LFSR shown in Figure 17.13. The cryptanalyst has thus learned the connections of the LFSR, together with the starting state of the register at time t1. He can therefore know the sequence for all time [3]. To generalize this example for any n-stage LFSR, we rewrite Equation (17.27) as follows:

We can write Equation (17.29) as the matrix equation

where

and

It can be shown [3] that the columns of X are linearly independent; thus X is non-singular (its determinant is nonzero) and has an inverse. Hence,

The matrix inversion requires at most on the order of n3 operations and is thus easily accomplished by computer for any reasonable value of n. For example, if n = 100, n3 = 106, and a computer with a 1- μs operation cycle would require 1 s for the inversion. The weakness of a LFSR is caused by the linearity of Equation (17.31). The use of nonlinear feedback in the shift register makes the cryptanalyst’s task much more difficult, if not computationally intractable.

### 17.4.3 Synchronous and Self-Synchronous Stream Encryption Systems

We can categorize stream encryption systems as either synchronous of self-synchronous. In the former, the key stream is generated independently of the message, so that a lost character during transmission necessitates a resynchronization of the transmission and receiver key generators. A synchronous stream cipher is shown in Figure 17.15. The starting state of the key generator is initialized with a known input, I0. The ciphertext is obtained by the modulo addition of the ith key character, ki, with the ith message character, m i. Such synchronous ciphers are generally designed to utilize confusion (see Section 17.3.1) but not diffusion. That is, the encryption of a character is not diffused over some block length of message. For this reason, synchronous stream ciphers do not exhibit error propagation.

In a self-synchronous stream cipher, each key character is derived from a fixed number, n, of the preceding ciphertext characters, giving rise to the name cipher feedback. In such a system, if a ciphertext character is lost during transmission, the error propagates forward for n characters, but the system resynchronizes itself after n correct ciphertext characters are received.

In Section 17.1.4 we looked at an example of cipher feedback in the Vigenere auto key cipher. We saw that the advantages of such a system are that (1) a nonrepeating key is generated, and (2) the statistics of the plaintext message are diffused throughout the ciphertext. However, the fact that the key was exposed in the ciphertext was a basic weakness. This problem can be eliminated by passing the ciphertext characters through a nonlinear block cipher to obtain the key characters. Figure 17.16 illustrates a shift register key generator operating in the cipher feedback mode. Each output ciphertext character, ci (formed by the modulo addition of the message character, mi, and the key character, ki), is fed back to the input of the shift register. As before, initialization is provided by a known input, I0. At each iteration, the output of the shift register is used as input to a (nonlinear) block encryption algorithm EB. The low-order output character from EB becomes the next key character, ki + 1, to be used with the next message character, mi + 1. Since, after the first few iterations, the input to the algorithm depends only on the ciphertext, the system is self-synchronizing.

• 🔖 Save To Your Account

### InformIT Promotional Mailings & Special Offers

I would like to receive exclusive offers and hear about products from InformIT and its family of brands. I can unsubscribe at any time.

## Overview

Pearson Education, Inc., 221 River Street, Hoboken, New Jersey 07030, (Pearson) presents this site to provide information about products and services that can be purchased through this site.

This privacy notice provides an overview of our commitment to privacy and describes how we collect, protect, use and share personal information collected through this site. Please note that other Pearson websites and online products and services have their own separate privacy policies.

## Collection and Use of Information

To conduct business and deliver products and services, Pearson collects and uses personal information in several ways in connection with this site, including:

### Questions and Inquiries

For inquiries and questions, we collect the inquiry or question, together with name, contact details (email address, phone number and mailing address) and any other additional information voluntarily submitted to us through a Contact Us form or an email. We use this information to address the inquiry and respond to the question.

### Online Store

For orders and purchases placed through our online store on this site, we collect order details, name, institution name and address (if applicable), email address, phone number, shipping and billing addresses, credit/debit card information, shipping options and any instructions. We use this information to complete transactions, fulfill orders, communicate with individuals placing orders or visiting the online store, and for related purposes.

### Surveys

Pearson may offer opportunities to provide feedback or participate in surveys, including surveys evaluating Pearson products, services or sites. Participation is voluntary. Pearson collects information requested in the survey questions and uses the information to evaluate, support, maintain and improve products, services or sites, develop new products and services, conduct educational research and for other purposes specified in the survey.

### Contests and Drawings

Occasionally, we may sponsor a contest or drawing. Participation is optional. Pearson collects name, contact information and other information specified on the entry form for the contest or drawing to conduct the contest or drawing. Pearson may collect additional personal information from the winners of a contest or drawing in order to award the prize and for tax reporting purposes, as required by law.

If you have elected to receive email newsletters or promotional mailings and special offers but want to unsubscribe, simply email information@informit.com.

### Service Announcements

On rare occasions it is necessary to send out a strictly service related announcement. For instance, if our service is temporarily suspended for maintenance we might send users an email. Generally, users may not opt-out of these communications, though they can deactivate their account information. However, these communications are not promotional in nature.

### Customer Service

We communicate with users on a regular basis to provide requested services and in regard to issues relating to their account we reply via email or phone in accordance with the users' wishes when a user submits their information through our Contact Us form.

## Other Collection and Use of Information

### Application and System Logs

Pearson automatically collects log data to help ensure the delivery, availability and security of this site. Log data may include technical information about how a user or visitor connected to this site, such as browser type, type of computer/device, operating system, internet service provider and IP address. We use this information for support purposes and to monitor the health of the site, identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents and appropriately scale computing resources.

### Web Analytics

Pearson may use third party web trend analytical services, including Google Analytics, to collect visitor information, such as IP addresses, browser types, referring pages, pages visited and time spent on a particular site. While these analytical services collect and report information on an anonymous basis, they may use cookies to gather web trend information. The information gathered may enable Pearson (but not the third party web trend services) to link information with application and system log data. Pearson uses this information for system administration and to identify problems, improve service, detect unauthorized access and fraudulent activity, prevent and respond to security incidents, appropriately scale computing resources and otherwise support and deliver this site and its services.

This site uses cookies and similar technologies to personalize content, measure traffic patterns, control security, track use and access of information on this site, and provide interest-based messages and advertising. Users can manage and block the use of cookies through their browser. Disabling or blocking certain cookies may limit the functionality of this site.

### Do Not Track

This site currently does not respond to Do Not Track signals.

## Security

Pearson uses appropriate physical, administrative and technical security measures to protect personal information from unauthorized access, use and disclosure.

## Children

This site is not directed to children under the age of 13.

## Marketing

Pearson may send or direct marketing communications to users, provided that

• Pearson will not use personal information collected or processed as a K-12 school service provider for the purpose of directed or targeted advertising.
• Such marketing is consistent with applicable law and Pearson's legal obligations.
• Pearson will not knowingly direct or send marketing communications to an individual who has expressed a preference not to receive marketing.
• Where required by applicable law, express or implied consent to marketing exists and has not been withdrawn.

Pearson may provide personal information to a third party service provider on a restricted basis to provide marketing solely on behalf of Pearson or an affiliate or customer for whom Pearson is a service provider. Marketing preferences may be changed at any time.

## Correcting/Updating Personal Information

If a user's personally identifiable information changes (such as your postal address or email address), we provide a way to correct or update that user's personal data provided to us. This can be done on the Account page. If a user no longer desires our service and desires to delete his or her account, please contact us at customer-service@informit.com and we will process the deletion of a user's account.

## Choice/Opt-out

Users can always make an informed choice as to whether they should proceed with certain services offered by InformIT. If you choose to remove yourself from our mailing list(s) simply visit the following page and uncheck any communication you no longer want to receive: www.informit.com/u.aspx.

## Sale of Personal Information

Pearson does not rent or sell personal information in exchange for any payment of money.

While Pearson does not sell personal information, as defined in Nevada law, Nevada residents may email a request for no sale of their personal information to NevadaDesignatedRequest@pearson.com.

## Supplemental Privacy Statement for California Residents

California residents should read our Supplemental privacy statement for California residents in conjunction with this Privacy Notice. The Supplemental privacy statement for California residents explains Pearson's commitment to comply with California law and applies to personal information of California residents collected in connection with this site and the Services.

## Sharing and Disclosure

Pearson may disclose personal information, as follows:

• As required by law.
• With the consent of the individual (or their parent, if the individual is a minor)
• In response to a subpoena, court order or legal process, to the extent permitted or required by law
• To protect the security and safety of individuals, data, assets and systems, consistent with applicable law
• In connection the sale, joint venture or other transfer of some or all of its company or assets, subject to the provisions of this Privacy Notice
• To investigate or address actual or suspected fraud or other illegal activities
• To exercise its legal rights, including enforcement of the Terms of Use for this site or another contract
• To affiliated Pearson companies and other companies and organizations who perform work for Pearson and are obligated to protect the privacy of personal information consistent with this Privacy Notice
• To a school, organization, company or government agency, where Pearson collects or processes the personal information in a school setting or on behalf of such organization, company or government agency.

This web site contains links to other sites. Please be aware that we are not responsible for the privacy practices of such other sites. We encourage our users to be aware when they leave our site and to read the privacy statements of each and every web site that collects Personal Information. This privacy statement applies solely to information collected by this web site.