Modulo Arithmetic

Kloker and Posen, 1988] Kloker, K. L. and Posen, M. P. (1988). Modulo arithmetic unit having arbitrary offset values. U.S. Patent 4,742,479. 

This chapter begins with an investigation of integer division and modulo arithmetic. We then explore check digit schemes that employ the number theoretic concepts we have developed. The chapter ends with an application of these concepts to cryptography. 

Description and Argumentation. Besides the days of the week, can you think of other sets of numbers used around us whose manipulation can usefully be described in terms of modulo arithmetic Give examples of this application of modulo arithmetic and explain why it would be beneficial in the situations that you have selected. 

Several number theoretic concepts (prime numbers, modulo arithmetic, relatively prime, etc.) were introduced in the first three sections of this chapter. Many of them are central to the creation and implementation of check digit schemes. They also can be applied in the area of cryptography to create some fairly sophisticated codes that are extremely difficult to break. The RSA Public-Key Cryptosystem, discussed in this section, is one such code. 

Both of these problems are addressed in the RSA public-key cryptosystem developed by R. L. Rivest, A. Shamir, and L. Adleman [19], (20). Although it is a fairly simple cipher, it is very hard to break. Moreover. Alice and Bob do not need to meet and agree on a cipher. The code is based on modulo arithmetic, and Alice, the sender of the message, only needs to know how to encipher a message, while Bob, the receiver, is the only person who needs to know how to decipher it. The cipher works as follows. 

To find 5, Bob first must find k such that = 1 (mod ru), or in this case such that 7 = 1 (mod 2592). He starts by computing (mod 2592) (mod 2592), (mod 2592), and so on with r 7, until he finds a number k such that (mod 2592) ss 1. Then s (mod 2592). The modulo arithmetic techniques introduced earlier help simplify this task ... 

If the resulting number C has fewer than four digits, enough zeros are added before the number to make it a total of four digits. Since this sequence contains seven numbers, let Ml = 0208, Ma = 1303, M i = 2426, A/4 = 0818, A/5 = 2600, Me = 2618, and A/7 — 1524. The modulo arithmetic techniques introduced earlier help simplify the calculations. Only the results of each calculation are given here ... 

Example 5J.6. Let g be an element o/Z with operation Given the work on modulo arithmetic done in Chapter 2, for any positive integer m,... 

In the foregoing discussion the properties of the incidence matrix and the cycle matrix were illustrated in terms of a cyclic digraph, but the results on the ranks of these matrices actually hold true for any connected digraph with N vertices. For an undirected graph, M and C contain only 0 and 1 (sometimes referred to as binary matrices), mathematical relations of identical form are obtained except that modulo 2 arithmetic2 is used instead of ordinary arithmetic. The ranks of M and defined in terms of modulo 2 arithmetic are JV — 1 and C, as before, and Eqs. (10) and (11) are modified to read... 

Modular arithmetic derives from the concept of congruence modulo m, written symbolically as... 

In an abstract sense, this computation is related to such everyday arithmetic functions as telling the time of day on a digital watch. When the watch tells the time, it does not say 240 minutes past noon. It says 4 o clock or 4 00. To express the time of day, the digital watch uses several kinds of modulos (or small measures) which have been used for centuries 60 minutes in an hour, 12 hours in the a.m. or p.m. of a day, and so on. If the watch says it is 4 00 in the afternoon, then, from one frame of reference, it has subtracted 480 minutes from the 720 minute period between 12 noon to midnight. What remains is 240 minutes past noon. That is, 720 - 480 = 240. The remainder, 240, can be divided evenly by the modulo 60 (and by other numbers which we will ignore). 

When we tell the time everyday, however, we do not use Gauss s terminology. Our clocks are aheady divided into modulos and we simply note the hour and how many minutes come before or after the hour. The importance of Gauss s congruence theory is that he created the formulas that allowed an immense variety of arithmetic actions to be performed based on different sets of numbers. 

Althaus E, Kruglov E, Weidenbach C (2009) Superposition modulo linear arithmetic sup(la). In Ghilardi S, Sebastian R (eds.) Frontiers of combining systems. Proceedings of 7th international symposium, pp 84-99, FroCoS 2009, Trento, September 16-18,2009. LNCS vol 5749, Springer, Berlin... 

For an arbitrary integer n > 2, we let C denote the cycle with n vertices. These vertices are labeled 1,.. .,n, with arithmetic operations on labels performed modulo n. The homotopy type of the independence complexes of cycles allows an easy description as well. 

The defining relationship would then be obtained by multiplying the generator by X5 and reducing aU powers by modulo 2 arithmetic. Therefore,... 

If this were a 2-level experiment, then one would stop at this point. However, for higher-level experiments, powers of the above equation must be taken and reduced with modulo I arithmetic. Therefore, squaring this defining relationship and reducing everything modulo 3 gives... 

Thus, my conclusion is that to describe safety case arguments, we need a formalism that includes quantification, uninterpieted predicates and constants, set theory, and arithmetic - but the theorem proving needs pushbutton automation only for the unquantified case. These capabilities are (a subset of) the capabihties of formahsms built on, or employing, SMT solvers (i.e., solvers for the problem of Satisfiabihty Modulo Theories) (Rushby 2006). Modem SMT solvers are very effective, often able to solve problems with hundreds of variables and thousands of constraints in seconds. They are the subject of an aimual competition, and this has driven very rapid improvement in both their performance and the range of theories over which they operate. 

Adjacent orbits of length n. Here there are several possibilities. The following possibilities using arithmetic modulo n for the indices are depicted in Figure 5. 

Griggio, A. A Practical Approach to Satisfiability Modulo Linear Integer Arithmetic. Journal on Satisfiability, Boolean Modeling and Computation (JSAT) 8, 1-27 (2012)... 

Lua supports the usual array of arithmetic operators + for addition, - for subtraction, for multiplication, / for division, % for modulo, for exponentiation and unitary - before any number. The supported relational operators are ... 

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