Implementing RSA Cryptography in Common Lisp

The RSA algorithm, while conceptually straightforward, presents several implementation challenges beyond the core mathematical operations. This guide walks through creating a functional RSA encryption library in Common Lisp, addressing the key hurdles encountered during development.

Core Algorithm Overview

The RSA algorithm relies on several mathematical steps:

1. Select two large prime numbers, p and q
2. Compute n = p × q, which forms part of both public and private keys
3. Calculate Euler's totient function φ(n) = (p-1)(q-1)
4. Choose a public exponent e (typically 65537)
5. Compute the private exponent d such that e × d ≡ 1 mod φ(n)

The encryption and decryption operations follow the same mathematical pattern:

Encryption: c = me mod n

Decryption: m = cd mod n

Implementation Structure

Our implementation will organize the RSA functionality into several components:

1. Key generation
2. Prime number generation
3. Modular arithmetic operations
4. Encryption/decryption core

Key Generation

The first step is creating our key structure and implementing the key generation process:

(defstruct rsa-key
  identifier bit-length key-type modulus exponent)

(defun generate-key-pair (identifier &optional (bit-size 2048))
  "Generate a public/private key pair for RSA encryption"
  (let* ((first-prime  (generate-large-prime (floor bit-size 2)))
         (second-prime (generate-large-prime (floor bit-size 2)))
         (modulus      (* first-prime second-prime))
         (public-exp   #x10001)
         (totient      (euler-totient first-prime second-prime))
         (private-exp  (modular-inverse public-exp totient)))
    (values (make-rsa-key :identifier identifier
                          :bit-length bit-size
                          :key-type :public
                          :modulus modulus
                          :exponent public-exp)
            (make-rsa-key :identifier identifier
                          :bit-length bit-size
                          :key-type :private
                          :modulus modulus
                          :exponent private-exp))))

(defun euler-totient (p q)
  "Calculate Euler's totient function for two primes"
  (* (- p 1) (- q 1)))

Encryption and Decryption Core

The encryption and decryption operations use the same mathematical function with different keys:

(defun transform-data (data-value key)
  "Encrypt or decrypt data using RSA with the provided key"
  (modular-exponentiation data-value 
                          (rsa-key-exponent key) 
                          (rsa-key-modulus key)))

Prime Number Generation

Generating large prime numbers efficiently is one of the main challenges. We'll implement the Miller-Rabin primality test:

(defun miller-rabin-test (n base)
  "Perform a single Miller-Rabin primality test"
  (= (modular-exponentiation base (- n 1) n) 1))

(defun is-prime (n)
  "Check if a number is prime using Miller-Rabin test"
  (or (= n 2)
      (and (> n 1)
           (oddp n)
           (every (lambda (a) (miller-rabin-test n a))
                  (generate-random-bases n 20)))))

(defun generate-random-bases (n count)
  "Generate a list of random bases for primality testing"
  (if (= count 0)
      nil
      (cons (+ 1 (random (- n 2)))
            (generate-random-bases n (- count 1))))))

(defun next-prime (starting-number)
  "Find the next prime number after the given value"
  (labels ((find-prime (current)
             (if (is-prime current)
                 current
                 (find-prime (+ current 2)))))
    (if (oddp starting-number)
        (find-prime starting-number)
        (find-prime (+ starting-number 1)))))

(defun generate-large-prime (&optional (bit-length 1024))
  "Generate a large prime number with specified bit length"
  (let ((hex-digits (floor bit-length 4))
        (random-hex (make-array 0 :element-type 'base-char 
                               :fill-pointer 0 :adjustable t)))
    (with-output-to-string (stream random-hex)
      (let ((high-bit (logior (random 16) #b1100))
            (*print-base* 16))
        (format stream "~A" high-bit)
        (dotimes (i (- hex-digits 1))
          (format stream "~A" (random 16)))))
    (next-prime (parse-integer random-hex :radix 16))))

Modular Arithmetic Operations

Two critical operations are modular exponentiation and computing modular inverses:

(defun modular-exponentiation (base exponent modulus)
  "Compute base^exponent mod efficiently"
  (cond ((= exponent 0) 1)
        ((evenp exponent)
         (mod (square (modular-exponentiation base (/ exponent 2) modulus)) 
              modulus))
        (t (mod (* base (modular-exponentiation base (- exponent 1) modulus)) 
                modulus))))

(defun extended-gcd (a b)
  "Compute the extended Euclidean algorithm"
  (if (= a 0)
      (values b 0 1)
      (multiple-value-bind (gcd x y)
          (extended-gcd (mod b a) a)
        (values gcd (- x (* (floor b a) y)) y))))

(defun modular-inverse (a m)
  "Compute the modular inverse of a modulo m"
  (multiple-value-bind (gcd x y)
      (extended-gcd a m)
    (declare (ignore y))
    (if (= gcd 1)
        (mod x m)
        (error "Modular inverse does not exist for given values"))))

Practical Considerations

While the core algorithm is now implemented, several practical challenges remain for a production-ready system:

1. Data chunking: Large messages must be split into blocks that fit within the key size
2. Padding schemes: Proper padding (like PKCS#1) is essential for security
3. Encoding/decoding: Converting between binary data and numbers for processing
4. Key management: Secure storage and handling of cryptographic keys

Additional cryptographic primitives like hash functions and symmetric encryption can complement the RSA implementation for a more comprehensive security solution.