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The numbers 7, 17, 19, 71, and 79 are primes, but how about 717-19-7179 Telling whether a reasonably large number is a prime seems tedious because there are far too many candidate factors to try However, there are some clever tricks to speed up the process For instance, you can omit even-valued candidates after you have eliminated the number 2 You can actually omit all candidates except those that are themselves primes In fact, a little further thought will convince you that you can proclaim N a prime as soon as you have rejected all candidates up to N , if N can indeed be factored as N = K L, for then it is impossible for both factors to exceed N We seem to be making progress! Perhaps by omitting more and more candidate factors, a truly ef cient primality test can be discovered Unfortunately, there is no fast primality test down this road The reason is that we have been trying to tell if a number is a prime by factoring it And factoring is a hard problem! Modern cryptography, as well as the balance of this chapter, is about the following important idea: factoring is hard and primality is easy We cannot factor large numbers, but we can easily test huge numbers for primality! (Presumably, if a number is composite, such a test will detect this without nding a factor) 30

FIGURE 1-4

(continued)

Is there some litmus test that will tell us whether a number is prime without actually trying to factor the number We place our hopes in a theorem from the year 1640 Fermat s little theorem If p is prime, then for every 1 a < p, ap 1 1 (mod p)

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In the earliest days of the internet web pages were primarily, a static, read-only medium used to share information The most interaction a user had with a web application was clicking a link Today, modern web applications have evolved to the point that there is an almost continuous twoway stream of information between a user s web browser and the application server I d like to emphasize almost continuous because it s important to understand the true nature of the HTTP Protocol as it relates to modern web applications HTTP is a stateless protocol as a persistent connection is not maintained between the client web browser and the HTTP server In contrast, traditional client-server applications maintain a persistent connection between the client and the server The actions of the HTTP protocol are often defined in terms of request, response I like to add a third term and describe it as request, response, disconnect The emphasis on disconnect helps developers understand the true nature of the technology they are working with Consider the impact of removing the network connection would have on a web application compared to a client-server application The stateful client-server application would know immediately if the

shown in boldface here:

12:

Proof Let S be the nonzero integers modulo p; that is, S = {1, 2, , p 1} Here s the crucial observation: the effect of multiplying these numbers by a (modulo p) is simply to permute them For instance, here s a picture of the case a = 3, p = 7:

set loopTimes to choose from list {"1", "2", "3", "4", "5", "6", "7", "8", "9", "10"} with prompt "Open this many TextEdit documents:"

FIGURE 12-7

1 2 3 4 5 6 1 2 3 4 5 6

connection were severed whereas the web application would have no knowledge of this action Obviously, the end user in the web scenario would know when they clicked a link or posted a form and received no response, but the HTTP Server would have no knowledge of this Now that we have a better understanding of the stateless nature of a web application, let s talk about how APEX maintains session state When a user requests a page from an APEX application, the APEX engine checks to see if this is the first request from this user in this browser session If it is the first request, APEX assigns a new, unique number for this session For brevity, we ll refer to this unique user session identifier as the Session ID When you are using an APEX application, including the APEX Development Environment, notice the third parameter in the colon delimited list in the URL is a long seemingly random number This number is also stored in a hidden form field on every page called pInstance APEX also passes variables, such as search criteria, as parameters in the URL when the operation is an HTTP GET One of the most convenient aspects of APEX for developers is that this session state is automatically managed for them in database tables Once a value is set for an APEX Page or Application Item, the value of this item is maintained for the duration of the user s session One simple, yet security-friendly benefit to this is realized when passing data between pages If you as a developer set the value of an item on page 1, you can refer to it on any other page in the application without the need to constantly pass it through the URL

5 Create a tell block for TextEdit, as shown in boldface here:

Let s carry this example a bit further From the picture, we can conclude {1, 2, , 6} = {3 1 mod 7, 3 2 mod 7, , 3 6 mod 7}

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