Question

Consider the "candidate prime" n 15841. a. Use Fermat's test ("for a random integer a with 1 < a < n-1: if an-1 1( mod n), then n is composite, else n is probably prime") to check the value of n, taking in turn a = 2, 3, 5, 7. b. Use the Solovay-Strassen test on n with bases a = 2, 3, 5 ("For a random integer a with -) < a < n-1: if ( c. Now use the Miller-Rabin test on this n, taking bases a = 2 then a = 3. 1 a(n-1)/2( mod n) then n is composite, else n is probably prime").

Answer 1

Solution: · n= 15841. Fermat's test is used to perform the primarily test . Take a random "a o n-1=15840 o Now for a-2, a15840 mod 15841-1, o Thus, a 1modn o Thus n can be possible prime o Now compute o a15840 mod 15841 315840 mod 15841-1 o That is modn. For a-3 o Thus n can be possible prime o Now compute o a15840 mod 15841 515840 mod 15841-1 o That is a1-1modn. For a-5 o Thus n can be possible prime

o Compute, a15840 mod 15841-715840 mod 15841- 6790 That is an-l #1 mod n . For a=7. Thus n is complex. o o Thus n is a complex number. b. a-2: o 0 o a-3: compute a 72 mod n-2920 mod 15841-1关a/n Thus the number is composite. o compute o ar-1% m od n 37920 m od 1 5841-1#a/n o Thus the number is composite a-5: o compute x-D a.72 mod n 57920 mod 15841 0 1#a/n o Thus the number is composite. All the three tests failed here.