Introduction

The Caesar cipher is one of the simplest and most widely known encryption techniques. It is a type of substitution cipher in which each letter in the plaintext is shifted a certain number of places down the alphabet. For example, with a shift of 1, A would be replaced by B, B would become C, and so on. The method is apparently named after Julius Caesar, who used it to communicate with his officials.

Implementation

The Caesar cipher is implemented in OCaml as follows:

let islower c =
  c >= 'a' && c <= 'z'

let isupper c =
  c >= 'A' && c <= 'Z'

let rot x str =
  let upchars = "ABCDEFGHIJKLMNOPQRSTUVWXYZ"
  and lowchars = "abcdefghijklmnopqrstuvwxyz" in
  let rec decal x =
    if x < 0 then decal (x + 26) else x
  in
  let x = (decal x) mod 26 in
  let decal_up = x - (int_of_char 'A')
  and decal_low = x - (int_of_char 'a') in
  String.map (fun c ->
    if islower c then
      let j = ((int_of_char c) + decal_low) mod 26 in
      lowchars.[j]
    else if isupper c then
      let j = ((int_of_char c) + decal_up) mod 26 in
      upchars.[j]
    else
      c
  ) str

The rot function takes an integer x and a string str as arguments. The integer x represents the number of positions to shift each letter in the string str. The function returns a new string that is the result of applying the Caesar cipher with the given shift.

Here is how to use the function.

let () =
  let key = 3 in
  let orig = "The five boxing wizards jump quickly" in
  let enciphered = rot key orig in
  print_endline enciphered;
  let deciphered = rot (- key) enciphered in
  print_endline deciphered;
  Printf.printf "equal: %b\n" (orig = deciphered)
;;

Step-by-Step Explanation

  1. The islower function checks if a given character is a lowercase letter.
  2. The isupper function checks if a given character is an uppercase letter.
  3. The rot function initializes two strings: upchars containing all uppercase letters and lowchars containing all lowercase letters.
  4. The decal function takes an integer x and returns the result of decrementing x by 26 until it is non-negative.
  5. The x variable is assigned the result of applying decal to x and then taking the modulus of 26.
  6. The decal_up variable is assigned the value of x minus the integer value of the character ‘A’.
  7. The decal_low variable is assigned the value of x minus the integer value of the character ‘a’.
  8. The String.map function applies a function to each character in the string str.
  9. For each character c in the string str, the function checks if it is a lowercase letter using the islower function.
  10. If c is a lowercase letter, the function calculates a new index j by adding the integer value of c to decal_low and taking the modulus of 26. The new character is then obtained by taking the j-th character of the lowchars string.
  11. If c is an uppercase letter, the function calculates a new index j by adding the integer value of c to decal_up and taking the modulus of 26. The new character is then obtained by taking the j-th character of the upchars string.
  12. If c is not a letter, the function returns c unchanged.
  13. The function returns the resulting string.

Complexity Analysis

The time complexity of the Caesar cipher algorithm is O(n), where n is the length of the input string str. This is because the String.map function applies a function to each character in the string str, and this operation takes O(1) time per character. Therefore, the overall time complexity is O(n).

The space complexity of the algorithm is also O(n), since the resulting string has the same length as the input string.

In terms of security, the Caesar cipher is very weak and easy to break, since there are only 26 possible shifts and an attacker can easily try all of them. Therefore, it is not suitable for use in modern cryptography, but it can still be used for simple purposes such as obfuscation or educational purposes.