Does it work to break the RSA modulus m=2480646863865156240312764602400149650804

Question

Does it work to break the RSA modulus m=2480646863865156240312764602400149650804447106891126179090516987413492462101441 by a brute force approach? How long does it take?

In this part, we will explore a vulnerability of the RSA system, known as the common modulus problem. Suppose that Alice and Kate use the same RSA modulus m as above, but with different encryption exponents Subscript[e, Alice]=17 and Subscript[e, Kate]=169. Bob wants to send the message x to both Alice and Kate. He first encrypts x using Kate's public exponent, and sends Kate the ciphertext 249081508444719456881769789206569419957026519534551194619251016972858511647493. Then he encrypts x using Alice's public exponent and sends the ciphertext 548038685778480218593900000000000000000 to Alice. If Eve intercepts both of the ciphertexts, explain how she can recover the plaintext without knowing either decryption exponent, then apply her method to find x.

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Explanation / Answer

According to question, we have modulus as
m = 2480646863865156240312764602400149650804447106891126179090516987413492462101441

Now if we apply brute-force attack to this modulus of RSA, then its not each to crack it. Here in this question value of m is very high because of it, its nearly impossible to crack it using brute-force. In RSA brute-force is applied by factoring the modulus, that is not very possible in this case because of high value of modulus.

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