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.
This phrase
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
cos
sin
x ! , sinxx cos x
x
Show that r is a group homomorphism from R with addition to a subgroup of GL2; R. What is its kernel?
21.168
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
1
So once again we have
1
1
bn,1=2 = 1 =
1
b
n
2
This completes the proof th
3. Sets
Sets and functions
Equivalence relations
3.1 Sets
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 .
Corollary: Eul
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