As you know, RSA encryption works by raising a plaintext block $p$ to some power $e$ modulo some composite modulus $n$:
$$c = p^e \pmod n$$
Now suppose you encounter textbook RSA encryption where $p$ and $e$ are so small that $p^e < n$. This is a disaster for Alice, because Eve can retrieve the plaintext by simply calculating the $e$-th root of $c$:
$$p = \sqrt[e]{c} \quad (\text{if }p^e < n)$$
On the other hand, if $p^e \ge n$, then $c$ will not be an exact $e$-th power. In this case, you could try adding integer multiples of $n$ to $c$ until you end up with a number that does work; i.e.,
$$p = \sqrt[e]{c + kn} \quad (\text{for some }k \in \mathbb{Z})$$
Note that this works without knowing the factors of $n$. In practical situations, RSA uses a much larger value of $e$ (usually 65537) and a much larger value of $p$ (through the use of padding), making this sort of attack impossible.
###A couple of examples:
Using the same modulus and exponent:
e = 3
n = c7d6bca5802235a56df4faf08d0fb4701e9e94b886a5c3d0c6441c92e3d3a0e6
d7549720d814961b53385dc1fbd28237b028a93c10d08b2d47bf31c8bd6f7aa8
062c6d7479a2e12734d17d851955de11382cf42137f1fc40839df2562b7a91dc
3b6751f6060ceb3090837d760b748997a43a01919a0c8f2c6b1bfb72653de70a
8ae197eb1c6780a75914cdacf9aca56af35059b728584e9bb284c4791e8cfed1
73ec641e7cc5b278ff680086b934c0abb2ec1f431b1d2173eac97019ff5100b0
b510fc87ff47a07e7c748f67c768d7dd739adfdd7d082e933d6038ca554f3282
d7aa9072028f4353dbef515b8eb6ff5867c039e1bd1f807a0cbed4655862060f
see if you can decipher the following:
(a) A simple example where $p^e < n$ (making the value of $n$ irrelevant):
c = 000608a5a8b32e3b218b49f9f0c33d05f135eff53dabee5ac00ec13e72997ef7
c4b478d6b95cecc01ebcf7cecf24785ba28b4b88f605ee93d6502f24635624cc
f4772e0d7334d125e047853c4a42e21d4bc3412729b0d230c1f5d6a2f8272bb1
8927313328c6e431bf1e38d5200a5ac32293f4d58c467636a20bc7393d7133c3
65880f07a2a29ad63fb21ebd2011b02dcd298fbd97c70d72fbc5c433a223987d
898da302aa100efd95f43e7d6698a7eeea05dd97870583f149beda3aa6334ba4
641c224e8c8f2cd61156abd42e4e3d03070c453656ec072526618e2ad74c87e8
9c54397de803b8e2347f7daf413695e1df816782758f26a2abfe947c82ce9e9d
(b) A slightly harder one where $p^e$ is larger than $n$. You'll have to test for multiple values of $k$ to crack this one:
c = 9478dff8ca00aa472291b24a2db6288c4a865cfb53bff7c6ecb304a2d506bbbd
16c4d26584657c17693becbdbba5a9bb18d462d72d52ae10bfe3edf906120bc2
b0bec55f529f8eb2a0ae6cd997db467c53c463d7ed44790c7917b76dc6992a4c
db7362e93eb3f4f2bb433d7256793a5992891a86ce87edc4234ec81e83c7ecc7
d7659fa3639a1e299745c8ef5bd632928cb4871f19a6bd63ec974e45ac132f04
f53f1e5c0e4a12d008783e31650816d2c78d9ec84ecd828a5ed7d915bf7a9a08
c23ebb85e69c9c2a3bef0d4a77b28a62db65f437e019062a74891ad17c3a9c58
ba73644a3097de31dbd5a13349f7a3f031d2352cbba680525e4249147aa58053
