In this question we show that we can use φ(n)/2. Let n = pq. Let
x be a number so that gcd(x, n) = 1.
1. show that xφ(n)/2 = 1 mod p and xφ(n)/2
= 1 mod q
2. Show that this implies that and xφ(n)/2 = 1 mod
n
3. Show that if e · d = 1 mod φ(n)/2 then xe·d = 1
mod n.
4. How can we use φ(n)/2 in the RSA?
Please explain answers...
1. Let α < β be real numbers and N ∈ N.
(a). Show that if β − α > N then there are at least N
distinct integers strictly between β and α.
(b). Show that if β > α are real numbers then there is a
rational number q ∈ Q such β > q > α.
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2. Let x, y, z be real numbers.The absolute value of x is
defined by
|x|= x, if x ≥...
3. (a) Let n ∈ N with n ≥ 2 and consider (Zn, ⊕). If a ∈ Zn,
show that a = a −1 if and only if n | 2a. What conditions on n
would guarantee that no element is equal to its own inverse?
(b) Let p be prime and consider (Up, ). Under what conditions
does a ∈ Up satisfy a = a −1 ? Can you specifically identify the
elements that satisfy this condition?
(a) Let n = 2k be an even integer. Show that x = rk
is an element of order 2 which commutes with every element of
Dn.
(b) Let n = 2k be an even integer. Show that x = rk
is the unique non-identity element which commutes with every
element of Dn.
(c) If n is an odd integer, show that the identity is the only
element of Dn which commutes with every element of
Dn.
(2) Let ωn := e2πi/n for n = 2,3,....
(a) Show that the n’th roots of unity (i.e. the solutions to zn
= 1) are
ωnk fork=0,1,...,n−1.
(b) Show that these sum to zero, i.e.
1+ω +ω2 +···+ωn−1 =0.nnn
(c) Let z◦ = r◦eiθ◦ be a given non-zero complex number. Show
that the n’th roots of z◦ are
c◦ωnk fork=0,1,...,n−1 where c◦ := √n r◦eiθ◦/n.
For all integers n > 2, show that the number of integer
partitions of n in which each part is greater than one is given by
p(n)-p(n-1), where p(n) is the number of integer partitions of
n.