CS 70 CHE ATSH EET ALEC LI
Note 1 (Propositional Logic)
•P=⇒ Q≡¬P∨Q
•P=⇒ Q≡¬Q=⇒ ¬P
•¬(P∧Q)≡¬P∨¬Q
•¬(P∨Q)≡¬P∧¬Q
•P∧(Q∨R)≡(P∧Q)∨(P∧R)
•P∨(Q∧R)≡(P∨Q)∧(P∨R)
•¬(∀xP (x))≡∃x¬P(x)
•¬(∃xP (x))≡∀x¬P(x)
Note 2/3 (Proofs)
• Direct proof
• Proof by contraposition
• Proof by cases
• Proof by induction
–Base case (prove smallest case is true)
–
Inductive hypothesis (assume
n=k
true for weak induction,
assume n≤ktrue for strong induction)
–Inductive step (prove n=k+1 is true)
• Pigeonhole principle
–Putting n+mballs in nbins =⇒≥1 bin has ≥2 balls
Note 4 (Sets)
•P(S)=powerset/set of all subsets; if |S|=k,|P(S)|=2k
• One to one (injection); f(x)=f(y)=⇒ x=y
• Onto (surjection); ¡∀y∃x¢¡f(x)=y¢; “hits" all of range
• Bijection: both injective and surjective
Note 5 (Countability & Computability)
• Countable if bijection to N
•
Cantor-Schroder-Bernstein Theorem: bijection between
A
and
B
if
there exists injections f:A→Band g:B→A
•
Cantor diagonalization: to prove uncountability, list out possibil-
ities, construct new possibility different from each listed at one
place (ex. reals ∈(0, 1), infinite binary strings, etc)
•A⊆Band Bis countable =⇒ Ais countable
•A⊇Band Ais uncountable =⇒ Bis uncountable
•
Infinite cartesian product sometimes countable (
∅×∅×···
), some-
times uncountable ({0,1}∞)
•
Halting Problem: can’t determine for every program whether it
halts (uncomputable)
• Reduction of TestHalt(P, x) to some task (here, TestTask)
–define inner function that does the task if and only if P(x) halts
–
call
TestTask
on the inner function and return the result in
TestHalt
Note 6 (Graph Theory)
•Knhas n(n−1)
2edges
• Handshaking lemma: total degree =2e
• Trees: (all must be true)
–connected & no cycles
–connected & has n−1 edges (n=|V|)
–connected & removing an edge disconnects the graph
–acyclic & adding an edge makes a cycle
• Hypercubes:
–n
-length bit strings, connected by an edge if differs by exactly 1
bit
–n
-dimensional hypercube has
n
2
n−1
edges, and is bipartite
(even vs odd parity bitstring)
•
Eulerian walk: visits each edge once; only possible if connected and
all even degree or exactly 2 odd degree
•
Eulerian tour: Eulerian walk but starts & ends at the same vertex;
only possible if all even degree and connected
• Planar graphs
–v+f=e+2
–Pf
i=1si=2ewhere si=number of sides of face i
–e≤3v−6 if planar (because si≥3)
–e≤2v−4 if planar for bipartite graphs (because si≥4)
–nonplanar if and only if the graph contains K5or K3,3
–all planar graphs can be colored with ≤4 colors
Note 7 (Modular Artithmetic)
•x−1(modular inverse) exists mod mif and only if gcd(x,m)=1
• Extended Euclidean Algorithm:
x y ¥x/y¦a b
35 12 2 −1 3
12 11 1 1 −1
11 1 11 0 1
1 0
gcd
start
answer –new a=old b
–new b=a−bjx
yk
–
if
gcd
(
x,y
)
=
1, then
a=
x−1mod y,b=y−1mod x
• Chinese Remainder Theorem:
–find bases bithat are ≡1 mod miand ≡0 mod mjfor j=i
→bi=ci(c−1
imod mi) where ci=Qi=jmj
–x≡Paibi(mod Qmi)
–solution is unique mod Qmi
–mimust be pairwise relatively prime in order to use CRT
Note 8 (RSA)
• Scheme: for primes p,q, find ecoprime to (p−1)( q−1)
–public key: N=p q and e
–private key: d=e−1mod (p−1)(q−1)
–encryption of message m:me(mod N)=y
–decryption of encrypted message y:yd(mod N)=m
•
Fermat’s Little Theorem (FLT):
xp≡x
(
mod p
), or
xp−1≡
1
(mod p) if xcoprime to p
•
Prime Number Theorem:
π
(
n
)
≥n
lnn
for
n≥
17, where
π
(
n
)
=
# of
primes ≤n
• Breaking RSA if we know d:
–
we know
de −
1
=k
(
p−
1)(
q−
1), where
k≤e
because
d<(p−1)(q−1)
–
so
de −1
k=pq −p−q+
1;
pq =N
, so we can find
p,q
because we
know d,e,k
Note 9 (Polynomials)
• Property 1: nonzero polynomial of degree dhas at most droots
•
Property 2:
d+
1 pairs of points (
xi
distinct) uniquely defines a
polynomial of degree at most d
• Lagrange Interpolation:
–∆i(x)=Qi=j
x−xj
xi−xj
–P(x)=Piyi∆i(x)
• Secret Sharing (normally under G F (p)):
–P(0) =secret, P(1),. . ., P(n) given to all people
–P
(
x
)
=
polynomial of degree
k−
1, where
k
people are needed to
get the secret
•
Rational Root Theorem: for
P
(
x
)
=anxn+···+a0
, the roots of
P
(
x
)
that are of the form p
qmust have p|a0,q|an
Note 10 (Error Correcting Codes)
•
Erasure Errors:
k
packets lost, message length
n
; need to send
n+k
packets because P(x) of degree n−1 needs npoints to define it
•
General Errors:
k
packets corrupted, message length
n
; send
n+
2
k
packets
• Berlekamp Welch:
–P(x) encodes message (degree n−1)
–E
(
x
) constructed so that roots are where the errors are (degree
k); coefficients unknown
–Q(x)=P(x)E(x) (degree n+k−1)
–
substitute all (
xi,ri
) into
Q
(
xi
)
=riE
(
xi
), make system of equa-
tions
–solve for coefficients; P(x)=Q(x)
E(x)
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