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# RSA Improved

# Assignment

> The standard RSA only uses 2 prime factors, way too insecure. Let’s improve it by using more factors! More is better, ...right?

<CodeGroup>
  ```python rsa_improved.py expandable lines theme={null}
  from Crypto.Util.number import getPrime
  import math

  # string to decimal
  pt = int(open('flag.txt','rb').read().hex(),16);

  primes = [];
  n = 1
  # euler totient
  phi = 1
  # public key
  e = 65537

  while math.log2(n) < 640:
  	primes.append(getPrime(32));
  	n *= (primes[-1]);
  	phi *= (primes[-1] - 1);

  # No duplicates
  assert(len(primes) == len(list(set(primes))));
  # private key
  d = pow(e,-1,phi);
  # cipher text
  ct = pow(pt,e,n);

  def decrypt(ct):
  	# decode ciphertext for plaintext
  	pt = pow(ct,d,n);
  	# convert decimal back to string
  	return bytes.fromhex(hex()[2:]).decode("utf-8");

  print("n = " + str(n));
  print("e = 65537");
  print("ct = " + str(ct));
  ```

  ```text out theme={null}
  n = 10588750243470683238253385410274703579658358849388292003988652883382013203466393057371661939626562904071765474423122767301289214711332944602077015274586262780328721640431549232327069314664449442016399
  e = 65537
  ct = 5995952936037255929781924635247478684210608634033130708312547257115162490907542249878843535087479397093661825467058312432783733583919194527896596274111265902276347768535338414466405501311805051241244
  ```
</CodeGroup>

# Solution