Programming Praxis

We have seen the Miller-Rabin and Baillie-Wagstaff primality tests in previous exercises. Today we look at a test developed by Robert M. Solovay and Volker Strassen in 1977. The test is no longer used, having been superseded by the other two tests, but was historically of great importance in the development of the RSA cryptosystem.

The test is based on Euler’s Criterion, which states that a^(p−1)/2 ≡ (a / p) (mod p) for any odd prime number p and any integer a on the range 2 to p − 1 with gcd(a, p) = 1, where (a / p) is the Legendre symbol. If a number n being tested is prime, the test will indicate that n is prime; if a number n being tested is composite, the test will indicate that it is either prime or composite with equal likelihood, a coin flip. Thus, the Solovay-Strassen test chooses k different witnesses a; if any of them indicate n is composite, then it must be composite, but if none of them indicate n is composite, it is presumed prime with odds 2^−k.

If you are willing to assume the truth of the Extended Riemann Hypothesis, the Solovay-Strassen test can be made an absolute proof of primality by testing all prime a in the range 2 to min(n − 1, log² n); if none of the a indicate the compositeness of n, then n is prime on the ERH.

The Solovay-Strassen test is no longer used, for three reasons: First, it takes more work to implement, as it involves computing the Jacobi symbol as well as modular exponentiation, whereas the Miller-Rabin test involves only modular exponentiation. Second, the Miller-Rabin test has a better error bound, k⁻⁴ compared to k⁻². Third, all Euler witnesses are also Miller-Rabin strong witnesses to the compositeness of n. The presence of the absolute proof of primality on the ERH is also not a bar to the use of the Miller-Rabin test, as there is a similar proof of primality based on Miller-Rabin strong witnesses.

Your task is to write two versions of the Solovay-Strassen primality test, one probabilistic and one on the ERH. 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.

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We studied a simple assembler for a hypothetical computer in the two previous exercises. Today we finish our look at the assembler by writing a program that produces neat listings, like this one for our sample program:

# print sum of input numbers (terminated by zero)

000:  03010        ld    zero # initialize sum to zero          
001:  04011        st    sum                                    
002:  01000   loop get        # read a number                   
003:  08007        jz    done # no more input if number is zero 
004:  05011        add   sum  # add input to accumulated sum    
005:  04011        st    sum  # store new value back in sum     
006:  09002        j     loop # go back and read another number 

007:  03011   done ld    sum  # print sum                       
008:  02000        put                                          
009:  10000        halt                                         

010:  00000   zero const 0                                      
011:  00000   sum  const

The listing shows the memory address in the first column, the contents of memory at the beginning of the program in the second column, the mnemonic label in the third column, the opcode in the fourth column, the object of the operation in the fifth column, and the comment in the sixth column.

Your task is to write a program that produces assembler listings. 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.

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In the previous exercise we built an assembler for a hypothetical computer. Today we write a simulator for that hypothetical computer. The basic idea is simple: Start with a program counter at location 0, either perform the action at the location and advance the program counter to the next location, or set the program counter to the object of a jump instruction.

Your task is to write the simulator. 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.

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In this exercise and the next one we will write a simple assembler for a hypothetical computer; we are following the assembler in the book The Awk Programming Langauge by Alfred V. Aho, Brian W. Kernighan and Peter J. Weinberger. Our computer has a memory of a thousand five-digit words, a single accumulator, and eleven opcodes:

An assembly-language program is a series of blank lines and statements consisting of up to four fields: The first field, if it exists, is a label; it must start at the first position on the line and may not start with a digit. The second field, which is mandatory, is the opcode; it follows the optional label and mandatory spaces. The third field, which is used only with some opcodes, is the object; if it is present, it follows the opcode and mandatory spaces. The fourth field, which is optional, is a comment; everything on the line following a hash-sign is ignored. Here is a sample assembly-language program that prints the sum of a series of integers entered by the user, with the end of input marked by a 0:

# print sum of input numbers (terminated by zero)

     ld    zero   # initialize sum to zero
     st    sum
loop get          # read a number
     jz    done   # no more input if number is zero
     add   sum    # add input to accumulated sum
     st    sum    # store new value back in sum
     j     loop   # go back and read another number

done ld    sum    # print sum
     put
     halt

zero const 0
sum  const

The contents of memory after loading the sample program are shown on the next page.

Your task is to write a program that assembles a program written in our simple assembly language and loads the program into memory; we’ll write a simulator for our hypothetical computer in the next exercise. 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.

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Programs that create graphs and charts are an important tool in many commercial and technical endeavors; for instance, a programmer might look at a graph of memory usage during a debugging run of his program to identify hot spots that need to be trimmed.

A simple type of graph is a scatter graph, sometimes called an x/y plot. The input is a list of points on an x,y-grid; the output is a picture with the points plotted on a grid. A simple input file appears on the next page.

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

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The only output functions provided by standard Scheme are display, for plain output, and write, for system-formatted output. That’s rather limiting. At the other extreme, Lisp provides the format function, which has more options than you can dream about (before I discarded it in favor of the HyperSpec, the spine of my printed copy of CLtL2 was broken in two places, format and loop). Many languages provide some kind of formatted output — does anyone remember PIC in COBOL? In today’s exercise we will implement the printf function popularized by C and used in several languages.

There are three functions in the printf family: (sprintf fmt expr …) returns a string formatted according to the specification given by format and containing the values expr …; (printf fmt expr …) displays a string formatted similarly to sprintf to the current output port, and (fprintf port fmt expr …) displays a string formatted similarly to sprintf to the indicated port. The fmt and port arguments are always required.

The fmt argument is a string that contains literal text, escape sequences, and specifications of how the expressions should be formatted. A specification consists of a literal percent-sign %, zero or more modifiers, and a single-character specifier. The single-character specifiers that we will support are:

c ascii character
d decimal integer
f floating-point number
o octal integer
s string
x hexadecimal integer
% literal percent sign

There should be as many expressions as there are format specifiers in the fmt string, except that a % literal percent sign specifier does not consume an expression. As many as four modifiers may appear between the literal percent sign % that starts a specifier and the single-character specifier that ends it:

- left-justify the expression in its field; if not given, the expression is right-justified
0 left-pad with leading zeros instead of spaces
width pad field to this width using spaces (or zeros)
.prec digits to right of decimal point, or maximum string length

The modifiers must appear in the order shown above. The width and prec arguments are unsigned decimal integers.

Escape sequences are introduced by a backslash; the following escape sequences are supported:

\b backspace
\f formfeed
\n newline
\r carriage return
\t horizontal tab
\ddd character with octal value ddd, where ddd is 1 to 3 digits between 0 and 7 inclusive
\c any other character c literally, for instance \\ for backslash or \" for quotation mark

Depending on the environment, a newline may (or may not) imply a carriage return, or vice-versa. Several examples are given on the next page.

Your task is to write a function that provides formatted output for your language, using the definition of printf given above or some other specification that is suitable for your language and your aspirations. 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.

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As penance for teasing you on April Fool’s Day, I have another exercise on factoring. Today we will simulate a machine invented by Derrick H Lehmer (the son of the famous father-son-daughter-in-law trio of mathematicians named Lehmer) that uses bicycle chains to factor integers. Unlike the previous exercise, this factoring method is real; in fact, a variant of this method was the best available method to factor integers from the mid-1930s to the mid-1960s, and Lehmer’s machine found many valuable factorizations for the Cunningham Project. We’ll discuss a simplified version of the machine today, as described by Uli Meyer, and the full version of the machine in a future exercise. I am unable to find a copyright-free picture of Lehmer’s machine, but if you’re interested you could visit the Computer History Museum, or see Lehmer himself describe the machine.

The machine is based on Fermat’s method of factoring by a difference of squares: if n = x² − y² then the factors of n are (x − y) and (x + y). Fermat’s method starts with y = ⌈ √n ⌉, computes y² − n, and stops when the result is a square.

Lehmer’s machine identifies likely squares by looking at quadratic residues. An integer a is a quadratic residue modulo n if there exists some x such that x² ≡ a (mod n). For a number y² − n to be a square in Fermat’s algorithm, it must be a quadratic residue to many small bases. Lehmer’s machine represents quadratic residues as “open” points on a bicycle chain, then spins many bicycle chains of different lengths along a common axis and stops when all are quadratic residues; in many but not all cases, such a number will will indicate a factor of n.

Uli Meyer gives a better description, shows a useful diagram, and provides photos of his Lego sieve machine.

Your task is to write a program that simulates Lehmer’s machine to factor integers. 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.

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A few days ago, an announcement was made in /r/crypto of a new factoring algorithm that runs in time O(log^4 n). A paper describes the algorithm. And even better, coding for the new algorithm is simple:

1)Set m ← n.
2) Express m as a list of its decimal digits.
3) Express each decimal digit as a list of binary bits.
4) Append the individual lists of binary bits.
5) Convert the resulting list of binary bits into a decimal number. Call it d.
6) If the greatest common divisor of n and d is between 1 and n, it is a factor of n. Report the factor and halt.
7) Set m ← d and go to Step 1.

Steps 1 through 4 form the digital code of a number. For example, 88837 forms the bit-list 1 0 0 0, 1 0 0 0, 1 0 0 0, 1 1, 1 1 1, where we used commas to delineate the decimal digits. That bit-list is 69919 in decimal, which is the digital code of 88837. To find a factor of 88837, compute successive digital codes until a gcd is greater than 1; the chain runs 88837, 69919, 54073, 2847, 2599, 5529, 2921, 333, at which point gcd(88837, 333) = 37 and we have found a factor: 88837 = 7⁴ × 37. Factoring 96577 is even faster: the digital code of 96577 is 40319, and gcd(96577, 40319) = 23, which is a factor of 96577 = 13 × 17 × 19 × 23. But trying to factor 65059 = 17 × 43 × 89 causes an infinite loop. The paper gives a heuristic solution to that problem, but we’ll stop there and accept that sometimes a factorization fails.

Your task is to write a function that factors by digital coding. 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.

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