CODEWITHSUNDEEP
calgorithmgmp
Bartek
Bartek

Reputation: 69

Finding multiplicative inverse of x in Z(n), extended Euclides algorithm

I'm trying to find multiplicative inverse of x using GMP library; I know there is a built-in function, but I want to write my own.
This is my extended Euclides algorithm implementation:

void extended_euclides(mpz_t r,mpz_t x,mpz_t y,mpz_t a, mpz_t b){
    mpz_t r_2,r_1,x_2,x_1,y_2,y_1,q,tmp;

    mpz_inits(r,x,y,NULL);

    mpz_init_set(r_2,a);
    mpz_init_set(r_1,b);
    mpz_init_set_ui(x_2,1);
    mpz_init_set_ui(y_2,0);
    mpz_init_set_ui(x_1,0);
    mpz_init_set_ui(y_1,1);
    mpz_init_set_ui(q,0);
    mpz_init_set_ui(tmp,0);

    do{
        mpz_fdiv_qr(q,r,r_2,r_1);   
        //x
        mpz_mul(tmp,q,x_1);
        mpz_sub(x,x_2,tmp);
        //y
        mpz_mul(tmp,q,y_1);
        mpz_sub(y,y_2,tmp);

        mpz_set(x_2,x_1);
        mpz_set(x_1,x);     

        mpz_set(y_2,y_1);
        mpz_set(y_1,y);

        mpz_set(r_2,r_1);
        mpz_set(r_1,r);
    }while(mpz_cmp_ui(r,0)!=0);

    mpz_set(r,r_1);
    mpz_set(x,x_2);
    mpz_set(y,y_2);

    mpz_clears(r_2,r_1,x_2,y_2,x_1,y_1,q,tmp,NULL);
}

It works fine for all small numbers and for some big numbers but not for all, and I don't known why. Example numbers for which it doesn't work:

a=99493485436357509294299436068793093643611893389896126764674829386592836165461754466092785338067969036756243799506670417432259164622123562781847156006846186608672621538507317131150760491084706497192710261706218845591564505899259562270249156644155861984060987885202877640033289062925176647874893491223532714128
b=202287573793610924311033969010234326099

if I change b to:

b=202287573793610924311033969010234326199

it works fine (I changed first 0 from right side to 1); the result I get is:

-26280231501456618600907242915048902345641123248519760433640466576442417888637174268721528225514196371138187569270563190841794774411834326405888357503240710494456394764379952360665884114850067939183395690214208147924280567331029828334399167395301049535292042342359035346464834873473183771024039179653285711685

The correct result, calculated by GMP function and checked by me in equation b*b^-1 ≡ 1 mod (a), is:

73213253934900890693392193153744191297970770141376366331034362810150418276824580197371257112553772665618056230236107226590464390210289236375958798503605476114216226774127364770484876376234638558009314571492010697667283938568229733935849989248854812448768945542843842293568454189451992876850854311570247002443

Upvotes: 1

Views: 401

Answers (1)

rcgldr
rcgldr

Reputation: 28808

If a and b are co-prime, extended Euclid algorithm finds x and y such that

x a + y b == 1

but either x or y may be negative. For y = inverse of b modulo a,

if y < 0 then y = y + a,

which will convert y to a proper value modulo a (note my prior comment).

The wiki example finds t = inverse of a modulo n, and has the same check:

if t < 0 then t = t + n

http://en.wikipedia.org/wiki/Extended_Euclidean_algorithm#Modular_integers

Upvotes: 1

Related Questions