Using Pari/GP for Cryptography

Here are some notes on the features of Pari/GP that I have found demonstrated in cryptography class. Don't forget that the entire manual and a tutorial are available on-line in pdf format.

Built-in functions

Most of these commands are documented on pages 46-54 and 62-72 of the user's guide.

add, subtract, multiply a and b a+b, a-b, a*b
integer quotient when a is divided by b a\b
integer remainder when a is divided by b (i.e. a mod b) a%b
a raised to the power b (see powermod below) a^b
assign a the value b a=b
compare a to b a<b, a>b, a<=b, a!=b a==b
a `and' b, a `or' b, `not' a a &&b, a||b, !a
convert a to an element of Z_n Mod(a,n)
convert an element x of Z_n to an integer (see note) lift(x)
binary expansion of a binary(a)
n^th bit of a bittest(a,n)
random integer between 0 and n-1 random(n)
highest power of p dividing a valuation(a,p)
greatest common divisor of a and b gcd(a,b) and bezout(a,b)
chinese remainder theorem applied to x and y (note) chinese(x,y)
φ(n) eulerphi(n)
factor the integer a factorint(a)
is p a prime integer? isprime(p)
first prime larger or smaller than a nextprime(a), prevprime(a)
n^th prime number prime(n)
discrete log of x to the base g (note) znlog(x,g)
multiplicative order of x in Z_n znorder(x)
find a primitive root modulo p znprimroot(p)
Legendre symbol of a over b kronecker(a,b)
define an elliptic curve E E=ellinit([0,0,0,a,b])
add/subtract z and w on elliptic curve E elladd(E,z,w) ellsub(E,z,w)
multiply z by k on elliptic curve E ellpow(E,z,k)
quit gp \q or quit

Other useful devices

Use ? to get help. For example ?bezout will give you information on the bezout function.
Use \l to start and stop a log file of your session.
Use # to turn on/off a timer. Use ## to find out the time of the last computation.
/* comments */
The sequence (1/a)%n will give the inverse of the integer a modulo n (assuming a and n are relatively prime).
you can define a powermod function with the command powermod(x,k,m)=lift(Mod(x,m)^k)
For some functions, the parameters (I've written them as x,y,g) need to be of the form Mod(a,n) since they are considered to be members of Z_n.
There is a description of programming tools (loops, if statements, etc.) on pages 137-143 of the user's guide.

add, subtract, multiply a and b	`a+b, a-b, a*b`
integer quotient when a is divided by b	`a\b`
integer remainder when a is divided by b (i.e. a mod b)	`a%b`
a raised to the power b (see powermod below)	`a^b`
assign a the value b	`a=b`
compare a to b	`a<b, a>b, a<=b, a!=b a==b`
a `and' b, a `or' b, `not' a	`a &&b, a\|\|b, !a`
convert a to an element of Z_n	`Mod(a,n)`
convert an element x of Z_n to an integer (see note)	`lift(x)`
binary expansion of a	`binary(a)`
n^th bit of a	`bittest(a,n)`
random integer between 0 and n-1	`random(n)`
highest power of p dividing a	`valuation(a,p)`
greatest common divisor of a and b	`gcd(a,b) and bezout(a,b)`
chinese remainder theorem applied to x and y (note)	`chinese(x,y)`
φ(n)	`eulerphi(n)`
factor the integer a	`factorint(a)`
is p a prime integer?	`isprime(p)`
first prime larger or smaller than a	`nextprime(a), prevprime(a)`
n^th prime number	`prime(n)`
discrete log of x to the base g (note)	`znlog(x,g)`
multiplicative order of x in Z_n	`znorder(x)`
find a primitive root modulo p	`znprimroot(p)`
Legendre symbol of a over b	`kronecker(a,b)`
define an elliptic curve E	`E=ellinit([0,0,0,a,b])`
add/subtract z and w on elliptic curve E	`elladd(E,z,w) ellsub(E,z,w)`
multiply z by k on elliptic curve E	`ellpow(E,z,k)`
quit gp	`\q` or `quit`