How can I tell whether two .NET DLLs are the same?

To compare two .dll files you can use ildasm or any other tool for geting IL code.
I have created a sample with embedded ildasm in the dll file so that you can use it on every machine. When we disassemble an assembly we check if the ildasm.exe file exists in the executing assembly folder and if not the file is extracted there from our dll file.
Using the ildasm file we get the IL code and save it to a temporary file.
Then we need to remove the following three rows:

MVID – as I wrote before this is a unique GUID generated with every build

Image Base (The image base tells us as to where the program will be loaded in memory by the Windows loader.) – this is different with every build as well

Time-date stamp – the time and date when the ildasm is run

So we read the temp file content, remove these rows using the regexes, and then save the file content to the same file.
Here is the Disassembler class:

using System;
using System.IO;
using System.Linq;
using System.Reflection;
using System.Diagnostics;
using System.Text.RegularExpressions;

namespace FileHasher
{
    public class Disassembler
    {
        public static Regex regexMVID = new Regex("//\\s*MVID\\:\\s*\\{[a-zA-Z0-9\\-]+\\}", RegexOptions.Multiline | RegexOptions.Compiled);
        public static Regex regexImageBase = new Regex("//\\s*Image\\s+base\\:\\s0x[0-9A-Fa-f]*", RegexOptions.Multiline | RegexOptions.Compiled);
        public static Regex regexTimeStamp = new Regex("//\\s*Time-date\\s+stamp\\:\\s*0x[0-9A-Fa-f]*", RegexOptions.Multiline | RegexOptions.Compiled);

        private static readonly Lazy<Assembly> currentAssembly = new Lazy<Assembly>(() =>
        {
            return MethodBase.GetCurrentMethod().DeclaringType.Assembly;
        });

        private static readonly Lazy<string> executingAssemblyPath = new Lazy<string>(() =>
        {
            return Path.GetDirectoryName(Assembly.GetExecutingAssembly().Location);
        });

        private static readonly Lazy<string> currentAssemblyFolder = new Lazy<string>(() =>
        {
            return Path.GetDirectoryName(currentAssembly.Value.Location);
        });

        private static readonly Lazy<string[]> arrResources = new Lazy<string[]>(() =>
        {
            return currentAssembly.Value.GetManifestResourceNames();
        });

        private const string ildasmArguments = "/all /text \"{0}\"";

        public static string ILDasmFileLocation
        {
            get
            {
                return Path.Combine(executingAssemblyPath.Value, "ildasm.exe");
            }
        }

        static Disassembler()
        {
            //extract the ildasm file to the executing assembly location
            ExtractFileToLocation("ildasm.exe", ILDasmFileLocation);
        }

        /// <summary>
        /// Saves the file from embedded resource to a given location.
        /// </summary>
        /// <param name="embeddedResourceName">Name of the embedded resource.</param>
        /// <param name="fileName">Name of the file.</param>
        protected static void SaveFileFromEmbeddedResource(string embeddedResourceName, string fileName)
        {
            if (File.Exists(fileName))
            {
                //the file already exists, we can add deletion here if we want to change the version of the 7zip
                return;
            }
            FileInfo fileInfoOutputFile = new FileInfo(fileName);

            using (FileStream streamToOutputFile = fileInfoOutputFile.OpenWrite())
            using (Stream streamToResourceFile = currentAssembly.Value.GetManifestResourceStream(embeddedResourceName))
            {
                const int size = 4096;
                byte[] bytes = new byte[4096];
                int numBytes;
                while ((numBytes = streamToResourceFile.Read(bytes, 0, size)) > 0)
                {
                    streamToOutputFile.Write(bytes, 0, numBytes);
                }

                streamToOutputFile.Close();
                streamToResourceFile.Close();
            }
        }

        /// <summary>
        /// Searches the embedded resource and extracts it to the given location.
        /// </summary>
        /// <param name="fileNameInDll">The file name in DLL.</param>
        /// <param name="outFileName">Name of the out file.</param>
        protected static void ExtractFileToLocation(string fileNameInDll, string outFileName)
        {
            string resourcePath = arrResources.Value.Where(resource => resource.EndsWith(fileNameInDll, StringComparison.InvariantCultureIgnoreCase)).FirstOrDefault();
            if (resourcePath == null)
            {
                throw new Exception(string.Format("Cannot find {0} in the embedded resources of {1}", fileNameInDll, currentAssembly.Value.FullName));
            }
            SaveFileFromEmbeddedResource(resourcePath, outFileName);
        }

        public static string GetDisassembledFile(string assemblyFilePath)
        {
            if (!File.Exists(assemblyFilePath))
            {
                throw new InvalidOperationException(string.Format("The file {0} does not exist!", assemblyFilePath));
            }

            string tempFileName = Path.GetTempFileName();
            var startInfo = new ProcessStartInfo(ILDasmFileLocation, string.Format(ildasmArguments, assemblyFilePath));
            startInfo.WindowStyle = ProcessWindowStyle.Hidden;
            startInfo.CreateNoWindow = true;
            startInfo.UseShellExecute = false;
            startInfo.RedirectStandardOutput = true;

            using (var process = System.Diagnostics.Process.Start(startInfo))
            {
                string output = process.StandardOutput.ReadToEnd();
                process.WaitForExit();

                if (process.ExitCode > 0)
                {
                    throw new InvalidOperationException(
                        string.Format("Generating IL code for file {0} failed with exit code - {1}. Log: {2}",
                        assemblyFilePath, process.ExitCode, output));
                }

                File.WriteAllText(tempFileName, output);
            }

            RemoveUnnededRows(tempFileName);
            return tempFileName;
        }

        private static void RemoveUnnededRows(string fileName)
        {
            string fileContent = File.ReadAllText(fileName);
            //remove MVID
            fileContent = regexMVID.Replace(fileContent, string.Empty);
            //remove Image Base
            fileContent = regexImageBase.Replace(fileContent, string.Empty);
            //remove Time Stamp
            fileContent = regexTimeStamp.Replace(fileContent, string.Empty);
            File.WriteAllText(fileName, fileContent);
        }

        public static string DisassembleFile(string assemblyFilePath)
        {
            string disassembledFile = GetDisassembledFile(assemblyFilePath);
            try
            {
                return File.ReadAllText(disassembledFile);
            }
            finally
            {
                if (File.Exists(disassembledFile))
                {
                    File.Delete(disassembledFile);
                }
            }
        }
    }
}

Now you can compare the contents of these two IL codes. Another option is to generate hash codes of these files and compare them. Hese is a HashCalculator class:
using System;
using System.IO;
using System.Reflection;

namespace FileHasher
{
    public class HashCalculator
    {
        public string FileName { get; private set; }

        public HashCalculator(string fileName)
        {
            this.FileName = fileName;
        }

        public string CalculateFileHash()
        {
            if (Path.GetExtension(this.FileName).Equals(".dll", System.StringComparison.InvariantCultureIgnoreCase)
                || Path.GetExtension(this.FileName).Equals(".exe", System.StringComparison.InvariantCultureIgnoreCase))
            {
                return GetAssemblyFileHash();
            }
            else
            {
                return GetFileHash();
            }
        }

        private string GetFileHash()
        {
            return CalculateHashFromStream(File.OpenRead(this.FileName));
        }

        private string GetAssemblyFileHash()
        {
            string tempFileName = null;
            try
            {
                //try to open the assembly to check if this is a .NET one
                var assembly = Assembly.LoadFile(this.FileName);
                tempFileName = Disassembler.GetDisassembledFile(this.FileName);
                return CalculateHashFromStream(File.OpenRead(tempFileName));
            }
            catch(BadImageFormatException)
            {
                return GetFileHash();
            }
            finally
            {
                if (File.Exists(tempFileName))
                {
                    File.Delete(tempFileName);
                }
            }
        }

        private string CalculateHashFromStream(Stream stream)
        {
            using (var readerSource = new System.IO.BufferedStream(stream, 1200000))
            {
                using (var md51 = new System.Security.Cryptography.MD5CryptoServiceProvider())
                {
                    md51.ComputeHash(readerSource);
                    return Convert.ToBase64String(md51.Hash);
                }
            }
        }
    }
}

You can find the full application source code on my blog here – Compare two dll files programmatically

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