Factor the Modulus

Public Modulus Is Visible

The tiny public modulus is enough to start trial division.

highlighted = computed this step

The public modulus is visible

The toy public key shows n=55 and e=3. The attack starts from those public values.

n=55,e=3n=55,\quad e=3
The public modulusTrial division rows are recomputed from the public toy RSA modulus.Try small divisors - n=55 factors to 5 x 11trialn mod trialdividesquotientstatus21no-no31no-no43no-no50yes11factor

The ciphertext is public too

The toy ciphertext is c=13. Factoring n is enough to rebuild the secret exponent.

c=13c=13
The public modulusTrial division rows are recomputed from the public toy RSA modulus.Try small divisors - n=55 factors to 5 x 11trialn mod trialdividesquotientstatus21no-no31no-no43no-no50yes11factor

Why factoring matters

If n splits into its prime factors, phi and d become ordinary recomputed arithmetic.

n=pqn=pq
The public modulusTrial division rows are recomputed from the public toy RSA modulus.Try small divisors - n=55 factors to 5 x 11trialn mod trialdividesquotientstatus21no-no31no-no43no-no50yes11factor

Summary

The toy public values are enough to start trial division because n is tiny.

trials=4\text{trials}=4
The public modulusTrial division rows are recomputed from the public toy RSA modulus.Try small divisors - n=55 factors to 5 x 11trialn mod trialdividesquotientstatus21no-no31no-no43no-no50yes11factor