REVISED: Thursday, February 8, 2024
1. INTRODUCTION
Prime numbers are very important for Internet security.
For example, a Bank could start with two large prime numbers p1 and p2. The Bank computer multiplies p1 * p2 and the result is used to create a Public Key to encrypt a Customers credit card.
The Bank keeps p1 and p2 secret. Therefore, p1 and p2 are referred to as Secret Keys.
The Secret Keys must be kept secret because they will be used to decrypt the credit card transactions received by the Bank. Once successfully decrypted the credit card transaction will access the Customer credit card bank balance and increase or decrease the Customer credit card account.
Prime Factorization using a tree is one decryption method:
It starts by dividing the Public Key by its least prime factor.
The example shown below does a Prime Factorization of the Public Key into two prime numbers which equal the Secret Keys.
21 -- Public Key
/\
3 7 -- Secret Keys are the unique Prime Factorization of the Public Key.
The above Public Key is a small number to make it easier to explain the relationship between the Public Key and the Secret Keys. In reality the Public Key would be a very large number that would take a very long time to determine the Prime Factorization even using a computer. Transactions would be completed before the Prime Factorizations would be computed.
2. HASKELL PROGRAM
The Haskell program is as follows:
-- C:\Users\Tinnel\Haskell2024\cryptography.hs
import Data.List
import System.IO
main :: IO () -- Has type I/O action.
main = do { putStrLn "" -- putStrLn Has type I/O action.
; putStrLn "=========================================================="
; putStrLn ""
; putStrLn "Pretend you are the Bank:"
; putStrLn ""
; putStrLn "Type a prime number for Secret Key p1 then press Enter."
; p1s <- getLine -- getLine is a function, getLine :: IO String
; putStrLn ""
; putStrLn "Type a prime number for Secret Key p2 then press Enter."
; p2s <- getLine
; putStrLn ""
; putStrLn "=========================================================="
; putStrLn ""
; putStrLn ("The Secret Keys p1 and p2 are " ++ p1s ++ " " ++ p2s ++ ".")
; putStrLn ""
; putStrLn "=========================================================="
; putStrLn ""
; let p1 = (read p1s)::Integer
; let p2 = (read p2s)::Integer
; let c = p1 * p2
; let cS = show c
; putStrLn ("The Public Key is (p1 * p2) which equals " ++ cS ++ ".")
; putStrLn ""
; putStrLn "=========================================================="
; putStrLn ""
; putStrLn "The Bank knows the Customer Secret Keys. Therefore,"
; putStrLn "the Bank decrypts the Customer Public Key with the Secret Keys."
; putStrLn ""
; putStrLn "For example, the Bank could compare the product of the Secret Keys with the credit card"
; putStrLn "transaction Public Key and if they equal the decryption is successful. Then the Bank can"
; putStrLn "process the credit card transaction to the correct Customer credit card account."
; putStrLn ""
; putStrLn "=========================================================="
; putStrLn ""
}
--
3. GHCi RUN
*Main> :l cryptography
[1 of 1] Compiling Main ( cryptography.hs, interpreted )
Ok, modules loaded: Main.
*Main> main
=========================================================================
Pretend you are the Bank:
Type a prime number for Secret Key p1 then press Enter.
3
Type a prime number for Secret Key p2 then press Enter.
7
=========================================================================
The Secret Keys p1 and p2 are 3 7.
=========================================================================
The Public Key is (p1 * p2) which equals 21.
=========================================================================
The Bank knows the Customer Secret Keys. Therefore,
the Bank decrypts the Customer Public Key with the Secret Keys.
For example, the Bank could compare the product of the Secret Keys with the credit card
transaction Public Key and if they equal the decryption is successful. Then the Bank can
process the credit card transaction to the correct Customer credit card account.
=========================================================================
*Main>
4. CONCLUSION
Using Secret Keys and a Public Key a Bank and a Customer can ensure secure credit card transactions.
5. REFERENCES
Bird, R. (2015). Thinking Functionally with Haskell. Cambridge, England: Cambridge University Press.
Davie, A. (1992). Introduction to Functional Programming Systems Using Haskell. Cambridge, England: Cambridge University Press.
Goerzen, J. & O'Sullivan, B. & Stewart, D. (2008). Real World Haskell. Sebastopol, CA: O'Reilly Media, Inc.
Hutton, G. (2007). Programming in Haskell. New York: Cambridge University Press.
Lipovača, M. (2011). Learn You a Haskell for Great Good!: A Beginner's Guide. San Francisco, CA: No Starch Press, Inc.
Thompson, S. (2011). The Craft of Functional Programming. Edinburgh Gate, Harlow, England: Pearson Education Limited.
