Toluu

Programming Praxis

A collection of etudes, updated weekly, for the education and enjoyment of the savvy programmer

Data Encryption Standard: Part 1August 31

The Data Encryption Standard has been one of the most successful ciphers in history, and is still in use today, especially in its Triple DES variant. The Data Encryption Standard is officially described by FIPS 46-3, though if you are not fond of reading algorithm descriptions written by government lawyers there are many other descriptions available on the internet.

DES is a block cipher, operating on 64 bits at a time. Here is an example:

PT  P  r  o  g  P  r  a  x
HEX 50 72 6F 67 50 72 61 78
KEY 01 23 45 67 89 AB CD EF
CT  CC 99 EA 46 B1 6E 28 90

There is more than one way to encrypt a message longer than 64 bits; we will examine them in a later exercise.

Your task is to write the code to encipher and decipher a single 64-bit block using the Data Encryption Standard. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

Chinese RemaindersAugust 27

In ancient China, two thousand years ago, a general wanted to count his troops. He first had them line up in ranks of eleven, and there were ten troops left over in the last rank. Then he had his troops line up in ranks of twelve, and there were four left over in the last rank. Finally he had them line up in ranks of thirteen, and there were twelve troops remaining in the last rank.

Your task is to determine how many troops the general had under his command. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.

Daniel Shanks’ Square Form Factorization AlgorithmAugust 24

In the mid-1970s, Daniel Shanks exploited the binary quadratic forms introduced by Legendre in the late eighteenth century and perfected by Gauss in the early nineteenth century to create a method of factoring integers. Binary quadratic forms are expressions of the form Ax2 + Bxy + Cy2 with A, B and C integers, normally represented as the triple (A, B, C). Binary quadratic forms are often called square forms.

We won’t discuss the mathematics of square forms, but we do need to know how to reduce a square form. A discriminant delta is calculated as Δ = b2 − 4ac. A single reduction step is called a rho reduction, and is given by ρ(a, b, c) = (c, r, (r2−Δ)/4c) where r is that unique integer b mod 2c such that −|c| < r <= |c|, if √Δ < |c|, or √Δ − 2|c| < r < √Δ, if |c| < √Δ.

A square form is in reduced form if |√Δ−2|a|| < b < √Δ; this reduction is accomplished by continually applying ρ until the reduced form is achieved.

Shanks’ method uses two loops. First, an initial square form is calculated, based on the number to be factored, which must be odd, composite, and not a perfect square; the details of that initialization are tedious, and appear in the solution on the next page. Then the square form is repeatedly reduced (this is the “forward” loop) until it is in proper form, which is described below. Then the reduced inverse square root of the square form calculated in the forward loop i

Marriage SortAugust 20

Today’s exercise is about a relatively new sorting algorithm. We start with an article Optimizing Your Wife by Kevin Brown, which proposes that the best way for a man to find a wife is to decide how many women he is willing to date before he chooses a wife, we’ll call that N, determine which of the first √N women is “best,” according to whatever matters to him, and then choose the next woman after the first √N that is better than any of the first √N women. For instance, to find the marriageable woman in a batch of a hundred, date ten of them, then marry the next one that is better than any of those ten. You may not find the optimal woman, but you’ll be close.

Eric Burnett turned Brown’s idea into a sorting algorithm. First, sample the first √N values at the beginning of an array, then swap any of the remaining values that are better than the greatest value of the sample to the end of the array, swap the greatest value of the sample just before those at the end, then recur on the smaller array before those greatest values. Finish the sort by performing insertion sort on the entire array; that will be quick, since most values are near their final positions.

Burnett’s algorithm requires three pointers: the current location of the end of the sample, the current location of the end of th

CutAugust 17

Unix V7 provided a utility called cut that reads its input a line at a time and selectively copies portions of each input line to standard output. Portions are selected either by character position or by character-delimited field. Cut is invoked as cut -clist [file ...] or cut -flist [-dchar] [file ...].

Character mode, invoked with the -c option, retains in the output those character positions that are mentioned in the list, which may contain column numbers, or ranges of column numbers separated by a dash, all separated by commas; counting starts from one. Field mode, invoked with the -f option, specifies a list of fields in a similar manner to character mode; fields are delimited by tab characters unless the field delimiter is changed by the -d option.

For example, the command cut -f1,3 -d: /etc/passwd prints user names and userid numbers from the password file.

Your task is to write a program to implement cut. When you are finished, you are welcome to read or run a suggested solution, or to post your own solution or discuss the exercise in the comments below.