#Please do not post homework
#Lucy Martinez, 04-17-2024, Assignment 24



# Problem 1: Decide whether you would like to sign up for the annual field trip
# to the Princeton cemetery followed by a free dinner.
# Due to the size of the class, there are two dates,
# each of them limited to eight people. 

# ANSWER: [6]
print([6]):





# Problem 2: Generalize
# AliceToBob(M,Nbob,eBob,Nalice,dAlice,H)
# and BobReadAlice(MS,Nbob,dBob,Nalice,eAlice,H) to

# AliceToBobG(ListM,Nbob,eBob,Nalice,dAlice,H)
# and BobReadAliceG(MS,Nbob,dBob,Nalice,eAlice,H)
# where ListM is an arbitrary list of integers each between 1 and Nbob,
# and the signature is the sum of the integers in List M cubed mod H.
# The encrypting and descryting is done separately for each member of ListM.
AliceToBobG:=proc(ListM,Nbob,eBob,Nalice,dAlice,H) local ListMsum,i,x,M1,S:
ListMsum:=add(i,i in ListM):

if ListMsum<=1 and ListMsum>=Nbob then
 print(`Sum of elements in ListM must be between 1 and Nbob`):
 return(FAIL):
fi:

M1:=[]:
for i from 1 to nops(ListM) do
 M1:=[op(M1),ListM[i]&^eBob mod Nbob]:
od:

x:=ListMsum&^3 mod H:
S:=x&^dAlice mod Nalice:
[M1,S]:
end:



BobReadAliceG:=proc(MS,Nbob,dBob,Nalice,eAlice,H) local i,Mlist,S,M,Msum,X,X1:
Mlist:=MS[1]: #this was M1
S:=MS[2]:
M:=[]:
for i from 1 to nops(Mlist) do
 M:=[op(M),Mlist[i]&^dBob mod Nbob]:
od:

Msum:=add(i,i in M):

X:=Msum&^3 mod H:
X1:=S&^eAlice mod Nalice:
if X<>X1 then
 print(`You are not Alice, I will call the police`):
fi:
M:
end:



# Problem 3: 
# Look at the class mailing list and find the three students right after you
# Email them your public ([N,e]) part of your public key (but keep d to yourself).
# Also email them your public H (so the Hash function is M^3 mod H)
# Ask them to email you THEIR [N,e] but to keep their d to themselves

# Convert your birthday to an integer: mm/dd/yyyy -> mmddyyyy.
# For example July 2, 1950 becomes the integer 02071950 (same as 2071950)
#  This is your secret message. Then for two of them sign them the right way,
#  but for one of them sign it the wrong way
# Ask them to send you back your birthday, and wether they believe it is you
# All together you should have six email exchanges
# Describe in detail all these interactions (on both ends),
# including the RSA keys and the Hash chosen



###################################C24.txt
Help24:=proc():
 print(`MakeRSAkey1(K) , MakeRSAkey(K,M) `): 
 print(`AliceToBob(M,Nbob,eBob,Nalice,dAlice,H)`):
 print(`BobReadAlice(MS,Nbob,dBob,Nalice,eAlice,H)`):
end:


with(numtheory):
#MakeRSAkey1(K): makes a triple [N,e,d] where N,e, are public and d private
#one try
MakeRSAkey1:=proc(K) local P,Q,ra,N,e,d:
ra:=rand(10^K..10^(K+1)):
P:=nextprime(ra()):
Q:=nextprime(ra()):
if P=Q then
 RETURN(FAIL):
fi:

N:=P*Q:
e:=rand(1..(P-1)*(Q-1))():

if gcd(e,(P-1)*(Q-1))<>1 then
  RETURN(FAIL):
fi:

d:=e&^(-1) mod (P-1)*(Q-1):
[N,e,d]:
end:



#MakeRSkey(K,M): tries to make an RSA key by trying M times
MakeRSAkey:=proc(K,M)  local m,rsa:
for m from 1 to M do
rsa:=MakeRSAkey1(K):
 if rsa<>FAIL then
   RETURN(rsa):
 fi:
od:
FAIL:
end:



#Hash function: x&^3 mod H
#AliceToBob(M,Nbob,eBob,Nalice,dAlice,H):
#Inputs Alice's message M, the [Nbob,eBob] public part of
#Bob's RSA key, the Nalice (public) and dAlice (private) of Alice's key
#and H (public Hash function x&^3 mod H)
#output a pair the encrypted message, followed by the much shorter
#signature
AliceToBob:=proc(M,Nbob,eBob,Nalice,dAlice,H) local x,M1,S:
M1:=M&^eBob mod Nbob:
x:=M&^3 mod H:
S:=x&^dAlice mod Nalice:
[M1,S]:
end:




#BobReadAlice(MS,Nbob,dBob,Nalice,eAlice,H)
#inputs a signed message MS=[M1,S] sent from Alice,
#Nbob,dBob,Nalice, eAlice, and H outputs
#the deciphered message from Alice and also checks
#that it came from Alice and not from a bad guy
#pretending to be Alice
BobReadAlice:=proc(MS,Nbob,dBob,Nalice,eAlice,H) local M1,S,M,X,X1:
M1:=MS[1]:
S:=MS[2]:
M:=M1&^dBob mod Nbob:
X:=M&^3 mod H:
X1:=S&^eAlice mod Nalice:
if X<>X1 then
 print(`You are not Alice, I will call the police`):
fi:
M:
end: