25.1 Ideals in commutative rings
The concept of ideal in a commutative ring is a sort of generalization of the concept of number. In fact,
originally there was a closely related notion of ideal number which extended the usual notion of number.
A subset T of a set S is a set all of whose elements are elements of S . This is written T S or S T .
So always S S and
S . If T S and T 6= and T 6= S , then T is a proper subset of S . Note that
the empty set is a subset of every set. For a subset T of a
Show that E is a group homomorphism from Q with addition to a subgroup of GL3; Q.
21.167 De ne a map r : R ! GL2; R by
x ! , sinxx cos x
Show that r is a group homomorphism from R with addition to a subgroup of GL2; R. What is its kernel?
Since in this situation she can make the encryption key public, often the encryption key e is called the
public key and the decryption key d is called the private key.
Elementary aspects of security of RSA
The security of RSA more or less depends upon the
And, therefore, in the case that M is even, the power of 2 dividing n , 1 is at least 2t+2. That is,
s t + 2. Therefore,
bn,1=2 = bm2s, = c2s, = c2t 2s, ,2t = ,12s, ,2t = 1 mod n
So once again we have
bn,1=2 = 1 =
This completes the proof th
Sets and functions
Here we review some relatively elementary but very important terminology and concepts about sets and
functions, in a slightly abstract setting. We use the word map as a synonym for function", as is
since 'p = p , 1 for prime p. By unique factorization in the ordinary integers, the only way that this
can happen is that ` be divisible by n, say ` = kn for some integer k. Then
y = g` = gkn = gk n mod p
That is, y is the nth power of gk .
This shows that b is a multiple of p. Done.
Corollary: of Lemma If a prime p divides a product a1a2 : : : an then necessarily p divides at least one of
the factors ai .
Proof: of Corollary This is by induction on n. The Lemma is the assertion for n = 2. S