My Arbitrary Precision Arithmetic Libraries

Hello C# code enjoyers. It's been a long time since I updated this blog, but that is going to change. I've been keeping a folder of blog post ideas and typed up thoughts on subjects that I think would fit here. So I have a lot of posts I wanted to make, about cool things I've written and discovered along the way.

For the coding side of things, boy have I been busy. I've written a number of arbitrary precision numeric type libraries, and other maths related projects.

I wanted to share some of these projects with you.

General Number Theory Sieve

The general number field sieve (GNFS) is currently the most efficient algorithm known for factoring very large semiprime numbers. It is the primary algorithm that has been used to set the last several world record factorizations.

• GNFS - This was a long, difficult project that took over a year to develop and required me teaching myself the basics of abstract algebra to be able to finish it. This project was intended to be more of a C# reference implementation of the General Number Field Sieve algorithm for the purpose of better understanding the General Number Field Sieve algorithm better than it was to be anything performant or fast.

Arbitrary Precision Arithmetic Types

• BigDecimal - An arbitrary precision, base-10 floating point number class. This is probably the most popular of my numeric libraries. It's getting quite popular, and may soon become the most downloaded 'big decimal' library on Nuget! One of the features my library has that its competitors do not is Trigonometric functions, Hyperbolic Trigonometric functions, and inverse trigonometric functions. Yes, to arbitrary precision. It accomplishes this by using the taylor series, and for the trig functions where the taylor series converges too slowly, ArcCosine for example, it uses continued fractions. The iteration of the infinite series stops once the difference between two iterations becomes less than some threshold you supply in terms of powers of negative 10, which is the same as saying how many digits past the decimal point do you want it to be accurate to.

• BigRational - Arbitrary precision number with arithmetic, except this one stores the value as an improper fraction under the hood. Actually, thats the fraction class of this library. BigRational represents the value as a mixed fraction. That is: Integer value + Fractional value

• BigComplex - Essentially the same thing as System.Numerics.Complex except that it uses a System.Numerics.BigInteger type for the real and imaginary parts instead of a double.

Polynomials

• Polynomial - The original. A univariate polynomial that uses System.Numerics.BigInteger for the type of the indeterminate (variables/letters).

• ComplexPolynomial - A univariate polynomial library that has System.Numerics.Complex type indeterminates.

And some variations using different underlying types for the variable:

Multivariate

• MultivariatePolynomial - A multivariate polynomial (meaning more than one indeterminate, e.g. 2XY^2) which uses BigInteger as the type for the indeterminates

• ComplexMultivariatePolynomial - A multivariate polynomial library that has System.Numerics.Complex indeterminates.

Generic Arithmetic

Seeing all these Polynomial libraries, you might be thinking: Wouldn't you rather make a class that you write only once and swap out the underlying type? Yes. Yes I would. Unfortunately, inheritance doesnt work. I thought generics would do it, but there is no generic constraints that says 'this is an arithmetic type. allow the +-*/ operators to work on these types in the normal way'. Thus the arithmetic operations are not available and thats kinda the whole point. 

Fortunately, there was a way to call each type's respective operator overload function dynamically at runtime by building a Linq.Expressions LambdaExpression that invokes the appropriate function on the class. This allows for you to specify the type as the generic type T with this generic class library just like you might have expected to be possible natively.

• GenericArithmetic - A core math library. Its a class of static methods that allows you to perform arithmetic on an arbitrary numeric type, represented by the generic type T, who's concrete type is decided by the caller. This is implemented using System.Linq.Expressions and reflection to resolve the type's static overloadable operator methods at runtime, so it works on all the .NET numeric types automagically, as well as any custom numeric type, provided it overloads the numeric operators and standard method names for other common functions (Min, Max, Abs, Sqrt, Parse, Sign, Log, Round, etc.). Every generic arithmetic class listed below takes a dependency on this class.

After writing that, naturally, I had to make:

• GenericPolynomial - A univariate polynomial library that allows the indeterminate to be of an arbitrary type, as long as said type implements operator overloading. This is implemented dynamically, at run time, calling the operator overload methods using Linq.Expressions and reflection.

• GenericMultivariatePolynomial - A multivariate polynomial that allows the indeterminates to be of [the same] arbitrary type. GenericMultivariatePolynomial is to MultivariatePolynomial what GenericPolynomial is to Polynomial, and indeed is implemented using the same strategy as GenericPolynomial (i.e. dynamic calling of the operator overload methods at runtime using Linq.Expressions and reflection).

• GenericVector - A generic Vector numeric type. Supports: Scalar arithmetic, vector arithmetic, square root, dot product, normal, reflection, distance, lerp, sum of squares and cosine similarity.

Miscellaneous

And some miscellaneous and smallish libraries playing around with various types of arithmetic:

• Continued Fraction - A continued fraction class. Arbitrary precision. Supports converting a continued fraction into rational approximations up to precision.

• IntervalArithmetic - Instead of representing a value as a single number, interval arithmetic represents each value as a mathematical interval, or range of possibilities, [a,b], and allows the standard arithmetic operations to be performed upon them too, adjusting or scaling the underlying interval range as appropriate. This is an interesting way of doing things and has some narrow uses. Despite that its sort of feels like a solution in need of a problem. See Wikipedia's article on Interval Arithmetic for further information.

The Self-Debugging-Code Design Pattern

Behold! The self-debugging-code design pattern:

try
{
  // Dodgy code here
}
catch (Exception ex)
{
  System.Diagnostics.Process.Start($"https://stackoverflow.com/search?q={System.Uri.EscapeDataString(ex.Message)}")?.WaitForExit();
}

I know, right? 🤯

Just replace the comment in the try block with something stolen from StackOverflow, and we've come full circle.

Obviously, this code is meant to only be used during development, and removed before going in production. Perhaps as a safe-guard (a poka yoke) against this winding up in production, one should put the code inside a conditional method and call it from within the catch block:

[Conditional("DEBUG")]
private static void DebugException(Exception exception)
{
  System.Diagnostics.Process.Start(  [...]
}

There; Now thats responsible developing!

Pascal's Triangle

Pascal's triangle has a lot of mathematically interesting properties; It represents binomial coefficients (n choose k or combination of a set), you can find within it the powers of 2, the powers of 11, the Fibonacci sequence, Sierpinski's triangle (a fractal), all the figurate numbers, Mersenne numbers and Catalan numbers, just to name a few. Furthermore, generating Pascal's triangle is quite simple, requiring only addition. Sometimes in mathematics something is both profound AND simple to understand. Such is the case with Pascal's triangle.

Recently, I made a contribution to rosettacode.org. If you have not checked out rosettacode.org before, you should definitely do so. I made a contribution to the Pascal's triangle task for the C# language. You can check it out here, or you can just view the code below.

My version of Pascal's Triangle is short, succinct, uses the BigInteger class for arbitrarily large numbers, and uses the algorithm to generate a single row of Pascal's triangle without needing to generate every row before it. This was originally the use case for writing this code in the first place; I wanted to generate high-numbered rows in a computationally feasible way. What does row # 5000 look like, for example? Well, could use the code below and generate the 5000th row of Pascal's triangle and format it as a string with the following one-liner: `string result = string.Join(" ", PascalsTriangle.GetRow(5000).Select(n => n.ToString()));`. You can also, of course, generate actual triangle-shaped rows of numbers:

                                                           1                                                           
                                                        1     1                                                        
                                                     1     2     1                                                     
                                                  1     3     3     1                                                  
                                               1     4     6     4     1                                               
                                            1     5    10    10     5     1                                            
                                         1     6    15    20    15     6     1                                         
                                      1     7    21    35    35    21     7     1                                      
                                   1     8    28    56    70    56    28     8     1                                   
                                1     9    36    84    126   126   84    36     9     1                                
                             1    10    45    120   210   252   210   120   45    10     1                             
                          1    11    55    165   330   462   462   330   165   55    11     1                          
                       1    12    66    220   495   792   924   792   495   220   66    12     1                       
                    1    13    78    286   715  1287  1716  1716  1287   715   286   78    13     1                    
                 1    14    91    364  1001  2002  3003  3432  3003  2002  1001   364   91    14     1                 
              1    15    105   455  1365  3003  5005  6435  6435  5005  3003  1365   455   105   15     1              
           1    16    120   560  1820  4368  8008  11440 12870 11440 8008  4368  1820   560   120   16     1           
        1    17    136   680  2380  6188  12376 19448 24310 24310 19448 12376 6188  2380   680   136   17     1        
     1    18    153   816  3060  8568  18564 31824 43758 48620 43758 31824 18564 8568  3060   816   153   18     1     
  1    19    171   969  3876  11628 27132 50388 75582 92378 92378 75582 50388 27132 11628 3876   969   171   19     1  

Awww, yes... that's pleasing.

Much of the pleasing effect is due to the CenterString(string, int) method (below).



And the code to do all that is thus:

public static class PascalsTriangle
{  
 public static IEnumerable GetTriangle(int quantityOfRows)
 {
  IEnumerable range = Enumerable.Range(0, quantityOfRows).Select(num => new BigInteger(num));
  return range.Select(num => GetRow(num).ToArray());
 }

 public static IEnumerable GetRow(BigInteger rowNumber)
 {
  BigInteger denominator = 1;
  BigInteger numerator = rowNumber;

  BigInteger currentValue = 1;
  for (BigInteger counter = 0; counter <= rowNumber; counter++)
  {
   yield return currentValue;
   currentValue = BigInteger.Multiply(currentValue, numerator--);
   currentValue = BigInteger.Divide(currentValue, denominator++);
  }
  yield break;
 }

 public static string FormatTriangleString(IEnumerable triangle)
 {
  int maxDigitWidth = triangle.Last().Max().ToString().Length;
  IEnumerable rows = triangle.Select(arr =>
    string.Join(" ", arr.Select(array => CenterString(array.ToString(), maxDigitWidth)) )
  );
  int maxRowWidth = rows.Last().Length;
  return string.Join(Environment.NewLine, rows.Select(row => CenterString(row, maxRowWidth)));
 }

 private static string CenterString(string text, int width)
 {
  int spaces = width - text.Length;
  int padLeft = (spaces / 2) + text.Length;
  return text.PadLeft(padLeft).PadRight(width);
 }
}

Note: This requires the System.Numerics library.

I make liberal use of Linq to keep the code short, yet expressive.

The code that generated the Pascal's triangle above, is:

IEnumerable triangle = PascalsTriangle.GetTriangle(20);
string output = PascalsTriangle.FormatTriangleString(triangle)
Console.WriteLine(output);

Other, arbitrarily large, number types available include the BigDecimal class, the BigComplex class, and the BigRational class. They are all available on my GitHub.


Thanks for stopping by!

EntropyGlance

Entropy at a glance

In a hurry? Skip straight to the C# source code - EntropyGlance; Entropy at a glance - A C# WinForms project - https://github.com/AdamWhiteHat/EntropyGlance



So I wrote an file entropy analysis tool for my friend, who works in infosec. Here it is, hands-down the coolest feature this tool offers is a System.Windows.Forms.DataVisualization.Charting visualization that graphs how the entropy changes across a whole file:

This application provides both Shannon (data) entropy and entropy as a compression ratio.
Get a more intuitive feel for the overall entropy at a glance with by visualizing both measures of entropy as a percentage of a progress bar, instead of just numbers.

   However, for those who love numbers, standard measures of entropy are also given as well. Information entropy is expressed both as the quantity of bits/byte (on a range from 0 to 8), and as the 'normalized' value (range 0 to 1). High entropy means it the data is random-looking, like encrypted or compressed information.
   The Shannon 'specific' entropy calculation makes no assumptions about the type of message it is measuring. What this means is that while a message consisting of only 2 symbols will get a very low entropy score of 0.9/8, a message of 52 symbols (the alphabet, as lower case first, then upper) repeated in the same sequence one hundred times would be yield a higher-than-average score of 6/8.
   This is precisely why I included a compression ratio as a ranking of entropy that is much closer to notion of entropy that takes into account repeated patterns or predictable sequences, in the sense of Shannon's source coding theorem.



Dive deep into the symbol distribution and analysis. This screen gives you the per-symbol entropy value and the ability to sort by rank, symbol, ASCII value, count, entropy, and hex value:

As always, the C# source code is being provided, hosted on my GitHub:
EntropyGlance; Entropy at a glance - A C# WinForms project - https://github.com/AdamWhiteHat/EntropyGlance

RC4 stream cipher variants and visualization of table permutation state

Here, I present some work I have been doing on two RC4 stream cipher variants. The first variant, as seen below, I wrote to help me visualize and understand what the RC4 tables was doing, and help me understand its properties.

Identity Permutation

The class that contains it is called SimpleTable and is exactly that; The simplest R4C implementation possible. It is notable in the fact that it does not use key scheduling at all, and its starting state is that of the Identity Permutation. The identity permutation is where the value at index zero equals zero, the value at index one is one, and so on. An easy way to remember what the Identity Permutation is, just recall the notion of a Multiplicative Identity (which is 1), where by multiplying a number N by the Multiplicative Identity gives you back the value of N, also known as the identity. Similarly, the Identity Permutation of an array A just gives you A. This is the trivial permutation. That is, there is no permuting of the array at all!

Anyways, this is done to see the perfectly ordered state, and how each round effects that state. In this way, we can visually check for the avalanche effect. In order to visualize the table, i just assign each value 0 to 255 a different shade of grey (I also have a rainbow-colored option that might be easier to tell apart similar values). At each step I create a Bitmap by looping through the table. Below, you can find an animated GIF of the first 100 steps of this cipher being applied to the identity permutation:


Notice how it takes a while to get going, and the first several values don't move much at all. After 256 steps, or one round, the cursor arrives back at index zero. Because the location of the first several values have not moved much or at all, we can clearly see that a mere 256 steps is insufficient at permuting the state enough to avoid leaking the first part of your key. Therefore it it is vital to permute the table for several rounds (256 steps per round) before you start using the stream.

Wired Equivalent Protocol

As some of you may know, WEP used RC4 with a weak key schedule. The key is spread out over 256 bytes using the following approach:

j = (j + table[i] + key[i mod keylength]) % 256;
SwapValues(table[i], table[j]);

and then it began streaming bytes from the table. Typically a nonce is concatenated to the key. Every time the table is set up and/or the nonce changes, some information about the key is leaked. Obviously a more secure procedure would use a hash of the key and the nonce, instead of the plain-text key, and to toss away the first 1024 bytes or so.

Cycle Length

Because each step in an RC4 cipher is a permutation, there is a limit to the number of unique bytes that can be produced before it begins repeating. This is called the permutation cycle. The length of the permutation cycle depends on the exact starting state, but we can get an upper bound.

Since there are 256 elements in the array, and two indices into the array (i and j), there is a maximum of
256! * 256^2 = 5.62 * 10^512 = 2^1700
possible states. That's 4.6 * 10^488 yottabytes!

This is the maximum possible states, however, and other starting states could have less. If the RC4 algorithm performed as a random permutation (which it does not, it performs worse), the cycle length would be half of the theoretical maximum above. Luckily the number above is so vast, that even some faction of it is still so many bytes that all of humanity has never and likely will never have that much total storage.

One thing to watch out for, however is something called Finney States. If an RC4 is started in one of these Finney States, the length of the cycle is much, much reduced. The chance of randomly generating one of these starting states, however, is VERY, very low.

Strengthening RC4

As stated, and visualized, above, it is vital to permute the table for several rounds (at 256 steps per round) after the key schedule, discarding the bytes, before you start using the stream. Also, it would be foolish to use the actual bytes of the key for permuting the starting state. It would be instead better to use a hash of the key + nonce or a key derivation function from the key instead the actual value of the key itself.

Another idea is, after shuffling the table enough rounds to hide the key, scramble the table an additional number of rounds, that value being some function of the key. This increases the possible starting states by whatever your range is.

In the classic RC4, each step would return one byte. The number of steps taken before returning each byte is configurable in my implementation.

Memory hardening

Check out the experimental branch for a memory hardened version. It stores the key class in memory, with the key XORed with a one-time pad, and then is protected in memory from access with the System.Security.Cryptography.ProtectedMemory class.

Other uses

The pseudo-random byte stream from the RC4 table is deterministic. Therefore if two remote computers with a shared secret, both computers can independently set up an RC4 table with exactly the same starting state and will get the same sequence of bytes which would be difficult to guess, given just the stream of bytes. If the plain text is XORed by the pseudorandom byte stream, then it can be decrypted by XORing it by the same byte steam.

The project includes 2 variants: 1) A simple table with a method to visualize the permutation state of the table and the avalanche effect as a bitmap 2) A more serious attempt at a secure implementation.


NOTE: THIS HAS NOT BEEN CRYPTO-ANALYZED AND PROBABLY NOT ACTUALLY SECURE, SO DO NOT TRUST IT!

Screenshots

Source code

Here is the GitHub page to the project (master branch).
Or just directly download the Zip file (experimental branch).

Lorenz Chaos Attractor

This project was inspired by one of Daniel Shiffman's 10 minute coding challenge YouTube videos, The Lorenz Attractor in Processing.

        dX = ((A * y)  -  (A * x)) * time;
        dY = ((B * x) -y -(x * z)) * time;
        dZ = ((x * y)  -  (c * z)) * time;

So it turns out this it not too terribly exciting. While its true that adjusting the starting values by a small amount change the behavior, if you go much outside the values its currently set for, you will end up with a pattern that quickly degenerates to a single, boring point. Personally, I was hoping for a more chaotic system. You might notice I am not using the 3rd point. I have yet to find a 3D drawing library that I like, though I need one for visualizing other projects. Anyways, since this was an experiment, I did the pragmatic thing and just made it 2D since I already knew how to do that.

Here is the result:

GitHub project

It wanted to draw the pattern very small, so I had to scale up the image by multiplying each number by some scale number.

One possibly useful idea is to use the cosine of the tangent of each number. This has the effect of canceling out the spiral and spreading the numbers out over a field. If you use just the tangent, you get a gradient from the top left corner. Perhaps you could use this as a pseudo-random noise source.

public static void TanCos(Lorenz system)
{
        system.x = 16 * (decimal)Math.Tan(Math.Cos((double)system.x));
        system.y = 16 * (decimal)Math.Tan(Math.Cos((double)system.y));
}

public static void Tan(Lorenz system)
{
        system.x = 6 * (decimal)Math.Tan((double)system.x);
        system.y = 6 * (decimal)Math.Tan((double)system.y);
}

Bloom Filter - A novel, space efficient data structure like a hash-table for billions of values.

Introduction

A bloom filter is a truly novel data structure. Similar to a hash table, it can tell you if you've hashed a particular value previously. You can add many, many more values to a bloom filter than you can to a hash table, does not degrade performance as the number of values in the set grows large, and requires only a fraction of the space of a hash table to store it!

This is not just an academic exercise, or something that only works in theory or in special cases. Indeed, companies like google use bloom filters to quickly determine if it has never seen that value before, thus avoiding a more costly lookup against a database every time the bloomfilter returns false.

Probabilistic

First off, its important to understand that a bloom filter is NOT a hash table, it operates in an entirely different way. A bloom-filter is what is known as a probabilistic data structure. What this means is, that it can tell you to within a certain probability, if an element exists in a set. In other words, false positive matches ARE possible, but false negative matches ARE NOT possible. For example, if you check a bloom filter for the existence of a value, and it returns false, you can know with 100% certainty that the bloom filter does not contain that value value before. However, if you test a value against the filter and it returns true, there is a small probability that it has in fact not seen that value before, but is returning a false positive. How big of a probability? Here's the beauty: It can be as small as you want it to be. It depends on a few factors, including the size of the filter, how full it is, and how many bits you use to store each value in the filter.

In a HashTable class, each item is stored as a key value pair, so the size of your object plus a 32 bit integer. Contrast that to a bloom filter, which stores only about 3-7 bits per value hashed. Also, my implementation applies compression when saving the filter to disk, providing even more space savings. A bloom filter with 160,000 values hashed and a 1% collision probability results in a filter that is 235KB uncompressed, and a whopping 54KB when compressed! Remember the filter is an array of bits. The entropy of the array is going to be at its greatest, and thus the compression ratio lowest, when exactly 1/2 of the bits are flipped, or the filter is half-way 'full'. This has the unusual property of getting smaller as you add more hashes to the filter. Actually, this is misleading--the actual filter itself never changes size, its only the compressed version that varies in size.

To handle the compression I just used the System.IO.Compression.DeflateStream class. An important note about working with this class: build an array of bytes and send your entire file in one go. In this way it will compress the whole file as one chunk. If you sent data to this stream piecemeal, it will compress each piece separately and you will get a poor compression ratio.

How it works

So how does this all work? The filter part of a bloom filter is just a large array of bits. You also require several different hash functions that each return a unique result for the same input value. When you add a value to the filter, the value is sent to about 3-7 different hash functions. Each hash function will return a value that is between 0 and the number of bits in the filter. Each value is used as an index to access and element on the array of bits that is your filter. When hashing a value, you just set the bit at each index location in the array to 1. Then testing for the presence of a value in the filter, you pass the value to the hash functions the same way as above, then visit each index, checking to see if any of them are 0. If even one bit at one of those index positions are zero, it means the filter has never seen that value before, because it would have set all those bits to 1. If all the bits at the index locations are 1, then it is likely that the filter has seen that value before. However,there is a chance that it is a false positive, because it could be that that value's different hashes all mapped to bits from other values. As the filter becomes more full, more bits are set to 1, and so the odds of a false positive go up. To build your filter by supplying the estimated number of values you think you are likely to store in the filter, and don't go above a certain ratio of 1 bits to 0 bits. If you were to let your filter hash so many values that every bit got set to 1, then the probability of receiving a false positive for a random value becomes 100%.

Solving the many hash problem

As I mentioned before, this requires several different hash functions that each return a unique result for the same input value. Although I said 3-7 hash functions, you might require 14 or more, when working with filters that can handle large number of hashes or a low false positive likelihood or both.

Instead of writing a bunch of separate hash algorithms, I implemented a stream cipher where in I just scramble the cipher table by a number of rounds that is unique to that input. Then, I can return as many indices as the filter is configured for. This sets up the table once per value. It needs to reset the table or else the indices that we mark will depend on every value that came before it, and in that particular order. Currently the bottle-neck is how many times it has the scramble the table for each value. If you need to hash really long values, you'll want to lower the number of rounds it scrambles the table.

Variations

In this implementation, the bloom-filter size is set once you create it, meaning that it cannot grow bigger if it gets too full, nor can you resize this bloom filter to become smaller if you sized it too big. Because multiple values could rely on the same bit, this implementation does not support removal of items, because to do so would cause several values to begin reporting false negatives.

In order to make a bloom filter that supports deletion, use a number like a byte instead of bits in your filter, and each time you visit an index in the filter while adding values, increment the number you find there. Then, to delete a value, visit each index as you did before, but decrement the number there. This way, if two values map to the same index, that information is tracked by incrementing the value. This is what is known as a Counting Bloom Filter.

There are other variants of bloom filters out there, including bloom filters that can grow in size if it gets too full, but such a thing is beyond the scope of my needs. In essence, when the filter gets too full, you create another separate filter, and add new values by first checking the first filter to see if it exists, and if not, adding the value to the second filter. Checking for the presence of a value requires checking both (and other) filters. For information on scalable bloom filters, please see this whitepaper.

The code

My C# Bloom Filter project on GitHub
Or download zip here.

Infix Notation Parser via Shunting-Yard Algorithm

Infix notation is the typical notation for writing equations in algebra.
An example would be: 7 - (2 * 5)

Parsing such an equation is not a trivial task, but I wanted one for my EquationFinder project, as I wanted to respect order of operations.

Strategies include substitution/replacement algorithms, recursion to parse into a tree and then tree traversal, or converting the infix notation to reverse polish notation (RPN), also known as post-fix notation, then using a stack based postfix notation evaluator. I choose the latter, as such algorithms are well defined in many places on the web.

My code consists of 3 classes, all static:
(Links go to the .cs file on GitHub)

1. InfixNotation - this simply holds a few public variables and calls the public methods on the below two classes.
2. ShuntingYardAlgorithm - this converts an equation in infix notation into postfix notation (aka RPN).
3. PostfixNotation - this evaluates the equation in postfix notation and returns a numerical result value.

In order to implement the shunting-yard algorithm and the postfix evaluator, I simply wrote the steps to the algorithms as written on Wikipedia:
(Links go to the Wikipedia article)
Link to the Shunting-Yard Algorithm to convert Infix notation to Postfix notation.
Link to the Postfix Notation Evaluation Algorithm.


The code for this is pretty extensive, but I will prettify it and present it below. Alternatively, you can view and download the code from the MathNotationConverter project on my GitHub.

InfixNotationParser:

public static class InfixNotation
{
   public static string Numbers = "0123456789";
   public static string Operators = "+-*/^";

   public static bool IsNumeric(string text)
   {
      return string.IsNullOrWhiteSpace(text) ? false : text.All(c => Numbers.Contains(c));
   }

  public static int Evaluate(string infixNotationString)
  {
    string postFixNotationString = ShuntingYardConverter.Convert(infixNotationString);
    int result = PostfixNotation.Evaluate(postFixNotationString);
    return result;
  }
}

ShuntingYardConverter 

(converts an equation from infix notation into postfix notation):

public static class ShuntingYardAlgorithm
{
   private static string AllowedCharacters = InfixNotation.Numbers + InfixNotation.Operators + "()";

   private enum Associativity
   {
      Left, Right
   }
   private static Dictionary<char, int> PrecedenceDictionary = new Dictionary<char, int>()
   {
      {'(', 0}, {')', 0},
      {'+', 1}, {'-', 1},
      {'*', 2}, {'/', 2},
      {'^', 3}
   };
   private static Dictionary<char, Associativity> AssociativityDictionary = new Dictionary<char, Associativity>()
   {
      {'+', Associativity.Left},
      {'-', Associativity.Left},
      {'*', Associativity.Left},
      {'/', Associativity.Left},
      {'^', Associativity.Right}
   };

   private static void AddToOutput(List<char> output, params char[] chars)
   {
      if (chars != null && chars.Length > 0)
      {
         foreach (char c in chars)
         {
            output.Add(c);
         }
         output.Add(' ');
      }
   }
   
   public static string Convert(string infixNotationString)
   {
      if (string.IsNullOrWhiteSpace(infixNotationString))
      {
         throw new ArgumentException("Argument infixNotationString must not be null, empty or whitespace.", "infixNotationString");
      }

      List<char> output = new List<char>();
      Stack<char> operatorStack = new Stack<char>();
      string sanitizedString = new string(infixNotationString.Where(c => AllowedCharacters.Contains(c)).ToArray());

      string number = string.Empty;
      List<string> enumerableInfixTokens = new List<string>();
      foreach (char c in sanitizedString)
      {
         if (InfixNotation.Operators.Contains(c) || "()".Contains(c))
         {
            if (number.Length > 0)
            {
               enumerableInfixTokens.Add(number);
               number = string.Empty;
            }
            enumerableInfixTokens.Add(c.ToString());
         }
         else if (InfixNotation.Numbers.Contains(c))
         {
            number += c.ToString();
         }
         else
         {
            throw new Exception(string.Format("Unexpected character '{0}'.", c));
         }
      }

      if (number.Length > 0)
      {
         enumerableInfixTokens.Add(number);
         number = string.Empty;
      }

      foreach (string token in enumerableInfixTokens)
      {
         if (InfixNotation.IsNumeric(token))
         {
            AddToOutput(output, token.ToArray());
         }
         else if (token.Length == 1)
         {
            char c = token[0];

            if (InfixNotation.Numbers.Contains(c)) // Numbers (operands)
            {
               AddToOutput(output, c);
            }
            else if (InfixNotation.Operators.Contains(c)) // Operators
               if (operatorStack.Count > 0)
               {
                  char o = operatorStack.Peek();
                  if ((AssociativityDictionary[c] == Associativity.Left &&
                     PrecedenceDictionary[c] <= PrecedenceDictionary[o])
                        ||
                     (AssociativityDictionary[c] == Associativity.Right &&
                     PrecedenceDictionary[c] < PrecedenceDictionary[o]))
                  {
                     AddToOutput(output, operatorStack.Pop());
                  }
               }
               operatorStack.Push(c);
            }
            else if (c == '(') // open brace
            {
               operatorStack.Push(c);
            }
            else if (c == ')') // close brace
            {
               bool leftParenthesisFound = false;
               while (operatorStack.Count > 0 )
               {
                  char o = operatorStack.Peek();
                  if (o != '(')
                  {
                     AddToOutput(output, operatorStack.Pop());
                  }
                  else
                  {
                     operatorStack.Pop();
                     leftParenthesisFound = true;
                     break;
                  }
               }

               if (!leftParenthesisFound)
               {
                  throw new FormatException("The algebraic string contains mismatched parentheses (missing a left parenthesis).");
               }
            }
            else // wtf?
            {
               throw new Exception(string.Format("Unrecognized character '{0}'.", c));
            }
         }
         else
         {
            throw new Exception(string.Format("String '{0}' is not numeric and has a length greater than 1.", token));
         }
      } // end foreach

      while (operatorStack.Count > 0)
      {
         char o = operatorStack.Pop();
         if (o == '(')
         {
            throw new FormatException("The algebraic string contains mismatched parentheses (extra left parenthesis).");
         }
         else if (o == ')')
         {
            throw new FormatException("The algebraic string contains mismatched parentheses (extra right parenthesis).");
         }
         else
         {
            AddToOutput(output, o);
         }
      }

      return new string(output.ToArray());
   }
}

PostfixNotation

(evaluates the postfix notation and returns a numerical result):

public static class PostfixNotation
{
   private static string AllowedCharacters = InfixNotation.Numbers + InfixNotation.Operators + " ";

   public static int Evaluate(string postfixNotationString)
   {
      if (string.IsNullOrWhiteSpace(postfixNotationString))
      {
         throw new ArgumentException("Argument postfixNotationString must not be null, empty or whitespace.", "postfixNotationString");
      }

      Stack<string> stack = new Stack<string>();
      string sanitizedString = new string(postfixNotationString.Where(c => AllowedCharacters.Contains(c)).ToArray());
      List<string> enumerablePostfixTokens = sanitizedString.Split(new char[] { ' ' }, StringSplitOptions.RemoveEmptyEntries).ToList();

      foreach (string token in enumerablePostfixTokens)
      {
         if (token.Length > 0)
         {
            if (token.Length > 1)
            {
               if (InfixNotation.IsNumeric(token))
               {
                  stack.Push(token);
               }
               else
               {
                  throw new Exception("Operators and operands must be separated by a space.");
               }
            }
            else
            {
               char tokenChar = token[0];

               if (InfixNotation.Numbers.Contains(tokenChar))
               {
                  stack.Push(tokenChar.ToString());
               }
               else if (InfixNotation.Operators.Contains(tokenChar))
               {
                  if (stack.Count < 2)
                  {
                     throw new FormatException("The algebraic string has not sufficient values in the expression for the number of operators.");
                  }

                  string r = stack.Pop();
                  string l = stack.Pop();

                  int rhs = int.MinValue;
                  int lhs = int.MinValue;

                  bool parseSuccess = int.TryParse(r, out rhs);
                  parseSuccess &= int.TryParse(l, out lhs);
                  parseSuccess &= (rhs != int.MinValue && lhs != int.MinValue);

                  if (!parseSuccess)
                  {
                     throw new Exception("Unable to parse valueStack characters to Int32.");
                  }

                  int value = int.MinValue;
                  if (tokenChar == '+')
                  {
                     value = lhs + rhs;
                  }
                  else if (tokenChar == '-')
                  {
                     value = lhs - rhs;
                  }
                  else if (tokenChar == '*')
                  {
                     value = lhs * rhs;
                  }
                  else if (tokenChar == '/')
                  {
                     value = lhs / rhs;
                  }
                  else if (tokenChar == '^')
                  {
                     value = (int)Math.Pow(lhs, rhs);
                  }

                  if (value != int.MinValue)
                  {
                     stack.Push(value.ToString());
                  }
                  else
                  {
                     throw new Exception("Value never got set.");
                  }
               }
               else
               {
                  throw new Exception(string.Format("Unrecognized character '{0}'.", tokenChar));
               }
            }
         }
         else
         {
            throw new Exception("Token length is less than one.");
         }
      }

      if (stack.Count == 1)
      {
         int result = 0;
         if (!int.TryParse(stack.Pop(), out result))
         {
            throw new Exception("Last value on stack could not be parsed into an integer.");
         }
         else
         {
            return result;
         }
      }
      else
      {
         throw new Exception("The input has too many values for the number of operators.");
      }

   } // method
} // class

Another alternative technique is to using the Shunting-Yard Algorithm to turn infix notation into an abstract syntax tree (Linq.Expressions anyone?). I will likely post this technique later.


Other blog posts by me that are related to this article are the Threaded Equation Finder, a Mixed Radix System Calulator and Drawing Text Along a Bezier Spline.

A Simple Word Prediction Library

   The word prediction feature on our phones are pretty handy and I've always and thought it would be fun to write one, and last night I decided to check that off my list. As usual, the whole project and all of its code is available to browse on GitHub. I talk more about the library and the design choices I made below the obnoxiously long image:

[Image of Windows Phone's Word Prediction feature]

   Visit the project and view the code on my GitHub, right here.
      (Project released under Creative Commons)

Overview:

   One thing you might notice, if for no other reason than I bring it up, is that I favor composition over inheritance. That is, my classes use a Dictionary internally, but they do not inherit from Dictionary. My word prediction library is not a minor variation or different flavor of the Dictionary class, and while it might be cool to access the word predictions for a word via an indexer, my word prediction library should not be treated as a dictionary.

Under the hood:

   There is a dictionary (a list of key/value pairs) of 'Word' objects. Each Word class has a value (the word), and its own dictionary of Word objects implemented as its own separate class (that does not inherit from Dictionary). This hidden dictionary inside each Word class keeps track of the probabilities of the the next word, for that given word. It does so by storing a Word as the key, and an integer counter value that gets incremented every time it tries to add a word to the dictionary that already exists (similar to my frequency dictionary, covered here).
The WordPredictionDictionary class doesn't grow exponentially, because each word is only represented once, by one Word class. The dictionaries inside the Word class only stores the references to the Word objects, not a copy of their values.
In order to begin using the WordPredictionDictionary to suggest words, one must train the WordPredictionDictionary on a representative body of text.

TODO:

• Write methods to serialize the trained data sets so they can be saved and reloaded. This has been implemented.
• Write an intelli-sense-like word suggestion program that implements the WordPredictionDictionary in an end-user application.

Thinq - A Linq Experiment

Thinq - A Linq Experiment

   View/Download the source code from the project's GitHub



   So I wrote a program, just an experiment, where I was making a range class using IEnumerables (C#), and each element doesn't have to increment by one, but any amount. so I was creating ranges like 7 to 10 million, increment by 7, so upon enumeration it would yield multiples of 7. This is also called arithmetic progression.

   Then I started combining different multiples with query operators like Where operator or Intersect like IEnumerable result = multiples7.Intersect(multiples13.MoveNext()), essentially creating a function that keeps only those numbers that are multiples of both 7 and 13, starting with the least common multiple.

   So I began testing. After some playing, I decided to take the first 7 primes, and find any common multiples to them between 1 and 10 million. Much to my surprise, it found all the common multiples of the first 7 prime numbers under 10 million (there are only two of them, 4849845 & 9699690), and it did it in 500 milliseconds on some very modest hardware (1 core, 2.16GHz, 4GB ram).

   I bumped up the ceiling to 50 million and I got an OutOfMemoryException because the IEnumerable holds on to every value it gets from the function MoveNext(). I threw in some metrics and discovered that it took about 3 seconds and some 32-million, 64-bit integers for my computer to declare 'out of memory'.

   Well, at least it was fast, even if it did eat up all my ram in 3 seconds, it was still promising. 

   The solution was to create an IEnumerator that was aware of the arithmetic sequences that constrained the results set. When MoveNext() is called repeatedly during enumeration, I avoid the infinite memory requirement by restricting the result set returned from MoveNext(); it returns the next whole number that is divisible by every arithmetic sequence's 'common difference', or increment value. In this way, you have created a enumerable sequence that is the _intersection_ of all of the sequences.

   The enumerator is prevented from running to infinity by obeying two limits: A maximum numeric value (cardinal) that GetNext() will return to ("results less than 50 million") and a maximum quantity of results (ordinal) that GetNext() yields ("the one millionth result").    If either of these limits are exceeded, the while loop will fail to evaluate to true. It is very common for my processor-intensive, long running or 'mathy' applications to employ a temporal limit (maximum time-to-live) or support cancellation, but this little experiment has been so performant that I have been able to get by without one.

   So what kind of improvement did we get out of our custom enumerable? I can now find all the common factors for the first 8 prime numbers up to 1 billion in 25 seconds! I was impressed; the application used to max out around 50 million and run out of memory, and now it can investigate to one billion in a reasonable amount of time and the memory it uses is not much more than the 8 or so integers in the result set. 1 billion, however seems to be the sweet spot for my single 2.13 GHz laptop. I ran the same 8 primes to 2 billion and it took 1 minute, 12 seconds:

TIME ELAPSED: 01:12.38
LCM[3,5,7,11,13,17,19,23] (max 2,000,000,000)

17 FACTORS:

111546435 
223092870 
334639305 
446185740 
557732175 
669278610 
780825045 
892371480 
1003917915 
1115464350 
1227010785 
1338557220 
1450103655 
1561650090 
1673196525 
1784742960 
1896289395

Mixed Radix Numeral System class and Counter

Mixed Radix Calculator

   My 'Mixed Radix Calculator' creates a counting system of radices (plural of radix), such as base 12 or mixed radices such as Minutes/Hours/Days/Years: 365:24:60:60. I choose the left side to be the most significant side. This is merely a personal preference, and my MixedRadixSystem class supports displaying both alignments.

   Of course you dont have to choose a mixed radix numeral system, you can count in an N-base numeral system, such as base 7 or a more familiar base 16. Another feature lies in my RadixNumeral class. Each numeral, or place value, supports having its own dictionary of symbols.

- View the code at its GitHub - https://github.com/AdamWhiteHat/MixedRadixCalculator

      (Project released under Creative Commons)

-  52:7:24:60:60:1000  -

  A numeral system (or system of numeration) is a writing system for expressing numbers.


  The most familiar one is of course the decimal numeral system. This is a 10-base numbering system. Computers use a binary numeral system. The base is sometimes called the radix or scale.

  Not all numbering systems have just one base. Take for example, how we currently divide time: There are 60 seconds in a minute, 60 minutes in an hour, 24 hours in a day, and 365 days in a year. This is called a mixed radix numeral system, and one might express the above mixed radix system like: 365:24:60:60.

  https://en.wikipedia.org/wiki/Mixed_radix
  http://mathworld.wolfram.com/Base.html

Uses:
  I haven't found a lot of use cases for it yet, but it is interesting. I originally built this because I wanted to experiment with numeral systems that uses increasing consecutive prime numbers for each radix, as well as experiment with some off-bases, such as base 3 or base 7.

  In a single base, say base 7, then 'round numbers' with only one place value having a 1 and the rest having zeros, such as 1:0:0:0:0 (in base 7), such numbers are powers of 7, and ever other number except for the 1's place value is a multiple of 7.

  A mixed radix numeral system can represent a polynomial, and possibly provide for a simpler way to visualize and reason about them.

  Yet another possible use is to make a numeral system with a base that is larger than and co-prime to some other target number (say 256) to make a bijective map from every value in a byte to some other value exactly once by repeatedly adding the value of the co-prime, modulus 256. This can appear rather random (or sometimes not at all) but the mapping is easily determined given the co-prime. I have talked about this notion before on my blog
  https://csharpcodewhisperer.blogspot.com/search/label/Coprime

  If you like this project you would probably like my project EquationFinder, it finds equations given constraints
  https://github.com/AdamWhiteHat/EquationFinder

Threaded Equation Finder

Threaded Equation Finder

Find arithmetic equations that equates to a given 'target' value, number of terms, and operators.

View the GitHub - Download source .zip

Introduction

   You should all be familiar with how a typical computer works; you give it some variables, describe some quantities of some resources you have, choose an algorithm, let it process, and it returns to you a result or outcome. Now imagine a computer if you could work with a computer that worked the other way around. I believe it was Douglas Adams that described the notion of an all-together different type of computer; That is, you tell the computer what you want the outcome to be, and it goes off figuring out how to get there and what you need to do it. Z3, the Theorem Prover, and the constraint satisfaction problem (CSP) solver (and probably others) in Microsoft's Solver Foundation do almost exactly that.
   There is also the idea of Backcasting, which is a similar, but different idea.

   My program isn't as fancy as all that, but it does find equations that equates to a given 'target' value, albeit at random. You define constraints other than just the target value, such as what operators are allowed in the equation, the quantity of terms there, and the range or set of allowed terms.
For example, how many different ways can 9 nines equal 27, using only addition, subtraction, and multiplication, if evaluated left-to-right (ignore the order of operations)? Turns out there are only 67 ways.

(above) Application Screen Shot

How it works

   The actual approach for finding equations that equate to an arbitrary target or 'goal' value is rather primitive. By way of Brute Force, several threads are launched asynchronously to generate thousands of random equations, evaluate them and keep the results that have not been found yet.

   This is something I wrote approx. 2 years ago. I dug it up and decided to publish it, because I thought it was interesting. As this was an 'experiment', I created different ways of storing and evaluating the expressions, and have made those different 'strategies' conform to a common interface so I could easily swap them out to compare the different strategies. I have refactored the code so that each class that implements IEquation is in its own project and creates its own assembly.

   There are two fully-working strategies for representing equations: one that represented the equation as a list of 2-tuples (Term,Operator), did not perform order of operations, and was rather obtuse. The other strategy was to store the equation as a string and evaluate it using MSScriptControl.ScriptControl to Eval the string as a line of VBScript. This was unsurprisingly slower but allowed for much more robust equation evaluation. Order of operations is respected with the ScriptControl strategy, and opens the way to using using parenthesis.

   The other idea for a strategy which I have not implemented but would like to, would be a left-recursive Linq.Expression builder. Also, maybe I could somehow use Microsoft Solver Foundation for a wiser equation generation strategy than at random.

Limitations

   Today, however, there are better architectures. A concurrent system like this would benefit greatly from the Actor model. If you wanted to write complex queries against the stream of equations being generated or selected or solved, maybe reactive extensions would be a slam dunk.

   Although this project certainly is no Z3, it does provide an example of an interface... perhaps even a good one.

Running on Raspberry Pi 2

   Microsoft's Managed Extensibility Framework (MEF) might be a good thing here, but I also wrote a console client that is designed to be ran with Mono on the Raspberry Pi 2. MEF is a proprietary Microsoft .NET technology that is not supported in Mono. The extra meta data in the assembly shouldn't be a problem, but having a dependency on the MEF assembly will be. Probing of the runtime environment and dynamically loading of assemblies is required here, which I have not had time to do, so at this time, there is no MEF.

   The reason the mono client is a console application is because mono and winforms on the Raspberry Pi 2 fails for some people. The problem has something to do with a hardware floating point vs a software float, and it happens to manifest itself when using a TextBox control. The only thing that I haven't tried, and that should presumably fix it, is to re-build mono from the latest source.

Certificate Enumerator

     Recently, my windows quit updating. Just prior to that, I had been messing around with my certificate store, so I suspected that to be the cause. Running Microsoft's troubleshooter reset the download, which I had a lot of hope of fixing the issue, but the download still continued to to fail. I decided to check the Windows Event Logs, and that's where I found an error message about a certificate in the chain failing. I knew it! However, I did not know whether a trusted certificate accidentally got put in the untrusted store, or whether an untrusted certificate was accidentally put in the trusted store. I needed a way to search all of my certificates' thumbprints or serial numbers against a know repository of trusted or untrusted certificates.
    Microsoft's certificate snap-in for MMC does not allow you to view certificates in an efficient way. Opening them one at a time, manually, and then scrolling all the way down to where the thumbprint is displayed to compare it to a webpage is painful. Also, I was not satisfied with the way that Microsoft allows you to search the certificate store. The search is not very effective and you cant even search for thumbprints! Also I do not believe the search feature allows you anyway to copy any of that information to clipboard.
Most of what I needed to accomplish could simply be done if I could just export all of my computers certificates thumbprint or serial numbers to a text file, csv file, or other simple and searchable format. Then I thought to myself, I know how to do that! It was the great the lack of features of the MMC certificate snap-in, and the inability to search for certificate thumbprints that inspired me to write my own certificate utility, known simply as Certificate Enumerator.




     CertificateEnumerator can list every certificate in your various certificate stores for your local machine and currently logged in user. It can then display that information to you either in a DataGridView or TextBox (as columnarized text), and provides the ability to persist that information to file as text, comma separated values (CSV), excel format or HTML table.


     The Certificate Enumerator also has the ability to 'validate' each certificate against its CRL (certificate revocation list), if it supplied one.


     The GUI could really use some love. In case you missed it, the project is on my GitHub, so feel free to download the source and play with it. If you come up with useful, submit a pull request.

Finding a date range in SQL

    At work, we use log4net, and we have the appender (or logging output location) set up to be AdoNetAppender, so thus it logs to a SQL database. In the Log table, there is a column called [Date], and it has the sql type of datetime.


    Often, when querying the Log table, you only want to view the most recent dates. lets say within the last week. You could always ORDER BY [Date] DESC of course, but suppose we wanted more control than that, such as only last week's dates.

    The SQL keywords (and functions) that are relevant here are BETWEEN, GETDATE and DATEADD.

    Here is the SQL code:

 SELECT
     [ID],[Date],[Thread],[Level],[Logger],[Message],[Exception]
 FROM
     [DatabaseName].[dbo].[Log]
 WHERE
     [Date] BETWEEN
      DATEADD(dd, -7, GETDATE())
      AND
      DATEADD(dd,  1, GETDATE())
 ORDER BY

     [Date] DESC

    The BETWEEN keyword should be pretty self-explanatory, as should the GETDATE function. The secret sauce here lies within the DATEADD function.

    The SQL function DATEADD has this signature: DATEADD (datepart, number, date) 

    The DATEADD function adds a number to a component of DATETIME, in this case, days. This number can be negative to subtract time from a DATETIME, as is the case with our example. The datepart parameter is what determines what component of the DATETIME we are adding to. You can add as much as a year, or as little as a nanosecond (what, no picoseconds? *laugh*). Microsoft's Transact-SQL MSDN page for DATEADD supplies the following table for datepart:

DATEPART	ABBREVIATIONS
year	yy, yyyy
quarter	qq, q
month	mm, m
dayofyear	dy, y
day	dd, d
week	wk, ww
weekday	dw, w
hour	hh
minute	mi, n
second	ss, s
millisecond	ms
microsecond	mcs
nanosecond	ns

    In the example, I am subtracting 7 days from the current date. If you are making a stored procedure, this variable can be replaced with a parameter:

 CREATE PROCEDURE [dbo].[sp_GetLogEntriesRange]
     @RangeInDays int
 AS
 BEGIN
     DECLARE @DaysToAdd int
     SET @DaysToAdd = (0 - @RangeInDays)

     SELECT
      [ID],[Date],[Thread],[Level],[Logger],[Message],[Exception]
     FROM
      [DatabaseName].[dbo].[Log]
     WHERE
      [Date] BETWEEN
       DATEADD(dd, @DaysToAdd, GETDATE())
       AND
       DATEADD(dd,  1, GETDATE())
     ORDER BY
      [Date] DESC
 END

    Enjoy, I hope this helps!