Séance no 3 : Autour de RSA 1 Présentation de RSA

Eric.Wegrzynowski (a) univ-lille1.fr
f:A→B
x∈A f(x)y=f(x)
y∈B
> 123456789098765445817313 *987654321012345680423311
121932631211705546391262024498778384392212583343
// Time 0
>ifactor (121932631211705546391262024498778384392212583343)
Evaluation time: 3.19
123456789098765445817313*987654321012345680423311
// Time 3.19
f(x) = xemod n x n
A=B=Z/nZf(x)
y=f(x)

m
n e
n p q
p= 17 q= 23 n= 391
>p:=17;q:=23;n:=p*q
17,23,391
e ϕ(n)=(p−1)(q−1)
ϕ(n) = 352 e= 3
> phi := (p - 1) *(q - 1); E := 3; igcd(E,phi)
352,3,1
m
memod n n −1
m= 21
> m := 21; c := irem(m^E,n)
21,268
c= 268
d
d=e−1mod ϕn,
d e Z/ϕ(n)Z
ed ϕ(n)d= 235
> d := irem(1/E,phi)
235
cdmod n
>irem(c^d,n)
21
m
c
m c =xemod n
n−1
n

m∈Z/nZ
c=memod n,
(n, e)c
Z/nZ
c
cdmod n,
d
n
m= 21
(n, e) = (391,3) irem
> c1 := m^E
9261
> c := irem(9261,391)
268
> m1 := c^d
4089560117124007355657007736693321034991773165026904638413048272\
0800981002743586222766326155798159208215524073816058399333380883\
4384196828144470725130828501536909280406518640876128180685816925\
0351277719188277596474711895159558519004569133385723126442099218\
5797845696054787887695873641521080669480895250118199474170802881\
8497920131116203565047945579665881953153361062259887760602654159\
5245494348238828886172913050474449650143745888778185461245859609\
9573533910284157482187242176850574398902255785475245635075199855\
33733186687853729012776947577894961992523366944441664274432
>irem(m1,391)
21

m1
>floor(log10(m1)) + 1
571
powmod abmod n
> n := 123456789098765445817313 *987654321012345680423311
121932631211705546391262024498778384392212583343
> a := 170980768765876465354365589709809
170980768765876465354365589709809
> b := 8768765465345576897098098098765765
8768765465345576897098098098765765
>powmod (a,b,n)
36701901377014928599189626904411362528587198281
ab
isprime
> isprime(2)
vrai
> isprime(3)
vrai
> isprime(4)
faux
rand k
k−1
kisprime vrai
faux

n := 100 + floor(rand (900)) ;
tantque not(isprime(n)) faire
n := 100 + floor(rand (900)) ;
ftantque ;
n
n
gen_premier t
t
n
p q
n=pq ϕ(n) = (p−1)(q−1) p6=q
n
e ϕ(n)
e
igcd
d
e ϕ(n)
irem 1/e ϕ(n)
(n, e)d
gen_rsa t
[n, e, d, p, q]n2t e
d p q n
isprime