#Store the ring of integers modulo 31 in a list called Z31
Z31 = IntegerModRing(31);
Z31;

#List the elements in Z31
list(Z31)

#Z31* is the subset of Z31 containing the elements of Z31 that are invertible modulo 31
Z31star = [x for x in Z31 if gcd(x,31)==1]; Z31star

#Output the number of elements in a given list
len(Z31);len(Z31star)

#Store the ring of integers modulo 45 in a list called Z45
Z45 = IntegerModRing(45);
list(Z45)

#Z45* is the subset of Z45 containing the elements of Z45 that are invertible modulo 45
Z45star = [x for x in Z45 if gcd(x,45)==1]; Z45star

#Output the number of elements in a given list
len(Z45);len(Z45star)

#The different patterns of (Z45,Z45*) and (Z31,Z31*) are due to the fact that
#31 is a prime number (is only divisible by 1 and itself)
#45 is not a prime number
#Let us test primality of a natural number
31 in Primes()

45 in Primes()

#Compute Euler's phi function... what does this function compute?
phi31 = euler_phi(31); phi45 = euler_phi(45);
phi31;phi45;

#In fact  phi45=phi9*phi5
phi5 = euler_phi(5); phi9 = euler_phi(9);
phi5;phi9;

#Modular reduction
Mod(17,4)

#Modular addition
Mod(17+25,4)

#Modular multiplication
Mod(17*25,4)

#Modular multiplication... more involved
Mod((2^1024+1)*(2^256),2^512) == 2^256

#A big prime!
p = 19557005198709002903944860293031210393112763219086296641657077101075696217230957575932867920574816565856404011416908876733063974554042055901449481509845537165833627083609136426386625960849573203936690977282752217830467255076903813033768202255463477317279507253420233052258146393806024179130344501759742626253484354878829682461413002143954786668386100202257237258007726886368228024987052808444032800050116408995869561195208576647050961019393809940446490709061546503439488751331252440431940049716566765873615456614911458658341732120931319694200689362617296630728389003180670326151653001322354906363854478585334821235883

#Number of bits
len(p.digits())

#Number of bits
len(p.bits())

#Binary representation of p
p.bits();

#Hexadecimal representation of an integer p
hexp=hex(p);
hexp

Integer(int(hexp,16));
p == Integer(int(hexp,16));

#From binary to integer representation
binp=bin(p);
Integer(int(binp,2));
p == Integer(int(binp,2));

#Computing SHA256 of a given character string - the output has 250 bits
import hashlib
dig = hashlib.sha256('This is an introduction to SageMath Cloud for Cryptography').hexdigest();
dig;
len(Integer(int(dig,16)).bits());

#Computing SHA256 of the string 'abc' - the output has 256 bits (as expected from SHA256)
abc = hashlib.sha256('abc').hexdigest();
abc;
len(Integer(int(abc,16)).bits());

#Computing SHA1 of the string 'abc' - the output has 160 bits (as expected from SHA160)
abcshort = hashlib.sha1('abc').hexdigest();
abcshort;
len(Integer(int(abcshort,16)).bits());