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RABiD BUNNY FEVER
K.T.K

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The math behind the RSA encryption algorithm

I’ve always thought that the RSA and Diffie–Hellman public key encryption algorithm systems are beautiful in their complex simplicity. While there are countless articles out there explaining how to implement them, I have never really found one that I think describes the math behind then in a simple way, so I thought I’d give a crack at it.

Both algorithms are derived from 3 math axioms:
  1. This is called Modular exponentiation (hereby referred to as modexp). In the following, x is a prime numbers and p is an integer less than x.
    1. p^(x  ) mod x = p (e.x. 12^(17  ) mod 17 = 12)
    2. p^(x-1) mod x = 1 (e.x. 12^(17-1) mod 17 = 1 )
  2. A further derivation from the above formulas shows that we can combine primes and they work in the same manner. In the following, x and y are prime numbers and p is an integer less than x*y.
    1. p^((x-1)*(y-1)  ) mod (x*y) = 1 (e.x. 12^((13-1)*(17-1)  ) mod (13*17) = 1 )
      Note: This formula is not used in RSA but it helps demonstrate how the formulas from part 1 becomes formula 2b.
      Due to how modexp works with primes, values of p that are multiples of x or y do not work with 2a.
    2. p^((x-1)*(y-1)+1) mod (x*y) = p (e.x. 12^((13-1)*(17-1)+1) mod (13*17) = 12)
  3. The final axiom is how modexp can be split apart the same way as in algebra where (x^a)^b === x^(a*b). For any integers p, x, y, and m:
    (p^(x*y) mod m) === ((p^x mod m)^y mod m)

With these 3 axioms we have everything we need to explain how RSA works. To execute an RSA exchange, encrypted from Bob and decrypted by Alice, the following things are needed.

The variable Variable name Who has it Who uses it Description
Prime Numbers 1 and 2 Prime1, Prime2 Alice Alice Alice will use these to derive variables PubKey, PrivKey, and Modulo. In our examples we use small numbers, but in reality, very large primes will be used, generally of at least 256 bit size.
Public key PubKey Alice, Bob Bob Alice sends this to Bob so he can encrypt data to her. Bob uses it as an exponent in a modexp.
Private key PrivKey Alice Alice Alice uses this to decrypt what Bob sends her. Alice uses it as an exponent in a modexp.
Modulo Modulo Bob, Alice Bob, Alice Alice sends this to Bob. They both use it as a modulo in a modexp
Payload Data Payload The data bob starts with and turns into EncryptedPayload. Alice derives Payload back from EncryptedPayload

Now, let’s start with axiom 2b:
Payload^((Prime1-1)*(Prime2-1)+1) mod (Prime1*Prime2) = Payload

Let’s change this up so the exponent is just 2 multiplications so we can use axiom 3 on it. We need to find 2 integers to become PubKey and PrivKey such that:
PubKey*PrivKey=(Prime1-1)*(Prime2-1)+1

And Modulo is Prime1*Prime2.
So we now have:
Payload^(PubKey*PrivKey) mod Modulo = Payload

Now, using axiom 3, we can turn it into this:
(Payload^PubKey mod Modulo)^PrivKey mod Modulo = Payload

Now, we can split this up into:
Bob calculates and sends to Alice: Payload^PubKey mod Modulo=EncryptedPayload
Alice uses the received EncryptedPayload and performs: EncryptedPayload^PrivKey mod Modulo = Payload

And the process is complete!


However, there is 1 caveat that I didn’t cover which makes the encryption that what we currently have weak. The calculation of PubKey and PrivKey from Prime1 and Prime2 needs to follow some rather specific complex rules to make the keys strong. Without this, an attacker may be able to figure out Prime1 and Prime2 from the Modulo and PubKey, and could then easily derive PrivKey from it. I generally see the PubKey as 65535, or another power of 2 minus 1.


My Tmux config

Tmux is a great alternative to gnu screen. I can’t believe I’ve never posted my custom Tmux config for Cygwin after all the work I put into it years ago. So here it is. Its features include:

  • Uses ctrl+a, like gnu screen, instead of ctrl+b
  • Mouse interaction is enabled
  • Tab bar/windows:
    • Current tab is highlighted in cyan
    • Cycle through tabbed windows with a click on its tab or ctrl+arrowkeys
    • Reorder tabbed windows with a drag of its tab or alt+arrowkeys
    • ctrl+a,/ to rename a tab on the tab bar
    • Create new window with ctrl+a,c
  • Panes
    • Create split panes with vertical=ctrl+a,| and horizontal=ctrl+a,-
    • Move around panes with click or ctrl+shift+arrowkeys
    • Resize panes by dragging on the separator bar or use ctrl+shift+alt+arrowkeys
    • Panes automatically resize to fit OS window
  • Clipboard/highlighting
    • Copy text to clipboard by highlighting it. Had to use a minor hack to fix a cygwin selection problem
    • Paste from clipboard with right click
    • Middle mouse button+drag starts copy mode
      • When in copy mode, u runs the selection as a command in a separate window (Instead of “cygstart” for cygwin, use “xdg-open” for linux, or “open” for MacOS X)
    • Double click selects word
    • Double middle click runs the word under the mouse as a command
  • Start the session on the current bash directory
  • Escape time is lowered for quicker response to scroll buffer access (ctrl+a,pageup)

To use this, save the file to ~/.tmux.conf

#Set current directory setting for cygwin
set-environment -g CHERE_INVOKING 1

#Mouse interaction
set -g mouse on

#Lower escape timing from 500ms to 50ms for quicker response to scroll-buffer access
set -s escape-time 50

#Window always takes up largest possible max size
set-window-option -g aggressive-resize

#Highlight active window in tab-bar at bottom in cyan
set-window-option -g window-status-current-bg cyan

#Reorder windows in status bar by drag & drop
bind-key -n MouseDrag1Status swap-window -t=

#Copy to clipboard on text selection in cygwin. Move cursor position 1 to the right before copy to bypass a bug
bind -Tcopy-mode MouseDragEnd1Pane send-keys -X cursor-right\; send -X copy-selection-and-cancel\; run-shell -b "tmux show-buffer > /dev/clipboard"

#Paste from clipboard with right click in cygwin
bind-key -n MouseDown3Pane run-shell 'tmux set-buffer -b winclip "$(cat /dev/clipboard)"'\; paste-buffer -db winclip

#Middle drag starts copy mode
bind -n MouseDrag2Pane copy-mode -M

#When in copy mode, "u" runs the selection as a command in a separate window (Instead of "cygstart" for cygwin, use "xdg-open" for linux, or "open" for MacOS X)
bind -Tcopy-mode u send -X copy-selection-and-cancel\; run-shell -b "tmux show-buffer | xargs cygstart"

#Double click selects word
bind-key -n DoubleClick1Pane copy-mode -M\; send-keys -X select-word

#Double middle click runs the word under the mouse as a command. See description for MouseDown3Pane above
bind-key -n DoubleClick2Pane copy-mode -M\; send-keys -X select-word\; send -X copy-selection-and-cancel\; run-shell -b "tmux show-buffer | xargs cygstart"

#Remap prefix to Control+a
set -g prefix C-a
unbind C-b
#bind 'C-a C-a' to type 'C-a'
bind C-a send-prefix

#Start in CWD when creating or splitting tabs; move the splitting planes keys to | and -
bind '|' split-window -h -c '#{pane_current_path}'  # Split panes horizontal
bind '-' split-window -v -c '#{pane_current_path}'  # Split panes vertically
bind c new-window -c '#{pane_current_path}' # Create new window
unbind '"'
unbind %

#prefix, / -- Renames window, but starts blank
bind-key / command-prompt "rename-window '%%'"

#Select next/prev window with Ctrl+(Left|Right)
bind-key -n C-Right next-window
bind-key -n C-Left previous-window

#Reorder window with Alt+(Left|Right)
bind-key -n M-Left swap-window -t -1
bind-key -n M-Right swap-window -t +1

#Switch panes using Ctrl+Shift+arrow
bind -n C-S-Left select-pane -L
bind -n C-S-Right select-pane -R
bind -n C-S-Up select-pane -U
bind -n C-S-Down select-pane -D

#Resize panes using Ctrl+Shift+Alt+arrow
bind-key -n C-S-M-Up resize-pane -U 1
bind-key -n C-S-M-Down resize-pane -D 1
bind-key -n C-S-M-Left resize-pane -L 1
bind-key -n C-S-M-Right resize-pane -R 1