Exercise 3: Beyond m=1 — Other Predictable Message Values — Possible Solution
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REASONING FROM FIRST PRINCIPLES: WHAT m^e mod n ACTUALLY COMPUTES
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Encryption is c = m^e mod n. The question for any specific m is
simply: does raising m to the power e, then reducing mod n, always
land on the same predictable value regardless of what e or n
actually are?

CASE 1: IS m=0 A PROBLEM?
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0^e = 0 for any positive exponent e (multiplying zero by itself
any number of times is still zero). So 0 mod n = 0 regardless of
n. This means m=0 ALWAYS encrypts to c=0, for any key at all -
exactly the same category of problem as this chapter's own m=1
finding. An attacker who sees a ciphertext of 0 knows immediately,
without needing the private key at all, that the original message
was 0.

CASE 2: IS m=n A PROBLEM?
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First, m=n isn't even a legitimately encodable message under this
system's own rules - valid messages are required to be in the
range [0, n-1], and m=n falls just outside that range. But suppose
it were used anyway: n mod n = 0, so m=n is congruent to 0 (mod n)
before the exponentiation even happens. Therefore n^e mod n = 0^e
mod n = 0 - it collapses to exactly the same degenerate case as
m=0, just reached by a different, technically-out-of-range route.

THE BROADER PATTERN
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Both cases share the same underlying cause: 0 and any exact
multiple of n (which includes n itself) map to a message that is
congruent to 0 (mod n), and 0 raised to any power is always 0,
regardless of which key is used. m=1 is a genuinely separate but
related case - not because it reduces to 0, but because 1 raised
to any power is always 1, for the same "trivial fixed point"
reason. The broader category is: any message value whose result
under exponentiation doesn't actually depend on the secret
exponent (0, 1, and by extension n-1 in some cases, since
(-1)^e alternates between 1 and -1 predictably) is a weak,
predictable message that real systems must avoid encrypting
directly.

RESULT
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Yes, m=0 is a genuine problem, for the same reason as m=1 -
0^e mod n = 0 always, leaking the message with zero effort. m=n is
technically out of the valid message range, but if it were
encrypted anyway, it would degrade to the exact same m=0 case,
since n ≡ 0 (mod n).

WHY THIS WORKS AS AN ANSWER
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Each case is reasoned through using the actual definition of
encryption (m^e mod n) rather than just pattern-matching to this
chapter's own m=1 example, and the underlying shared cause (a
message value whose exponentiation result doesn't depend on the
actual secret key) is identified explicitly, generalizing beyond
the single worked example rather than treating m=1 as an isolated
curiosity.