rsa lesson added

training/h4tum/Intro_to_RSA/challenge/Fermat/1.py

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+from Crypto.Util.number import inverse, getPrime, bytes_to_long, long_to_bytes
+from sympy import nextprime
+import os
+import math
+
+
+flag = "h4tum{...}"
+m = bytes_to_long(flag.encode())
+
+p : int = getPrime(512)
+diff = int.from_bytes(os.urandom(16), 'big')
+q : int = nextprime(p - diff)
+n = p * q
+phi = (p - 1) * (q - 1)
+e = 65537
+d = inverse(e, phi)
+
+ciphertext = pow(m, e, n)
+
+print(f"Here is the public key:\nn = {n}\ne = {e}")
+print(f"\nHere is the encrypted flag:\n{ciphertext}")
+
+t = math.isqrt(n)
+
+for i in range((n+1)//2 - t + 1):
+    val = (t + i)**2 - n
+    if val > 0:
+        root = math.isqrt(val)
+        if root * root == val:
+            r = math.isqrt((t+i)**2 - n)
+            rec_p = (t+i) - r 
+            rec_q = (t+i) + r
+            rec_phi = (rec_p-1) * (rec_q-1)
+            rec_d = inverse(e, rec_phi)
+            plain_text = pow(ciphertext, rec_d, n)
+            print("We recovered d yay")
+            print(f"Cracked flag: {long_to_bytes(plain_text)}")
+            break
+
+
+
+
+# Here is the public key:
+# Here is the encrypted flag:
+# 13158607848927768767396122915610010628021182023103983078656746030071489695567187651581550866461152226282740548654306454938594539749024910072164328096542525661951926739718955208177780816184581302682602611753650161772357698132248994074563798067850345731371433620586869504864647385937063506440611160969011596274