QUIZ CMPE-553 25.05.2005 (110 min, 2 points) St. Name, Surname______________________________________ St.Id#_____________ Open books, notes!! Open books, notes!! Open books, notes!! Open books, notes!! Instructor Alexander Chefranov Task 1. (0.2 points) Show that group (M, *) (where M is a set of square 2*2 matrices, * is a matrix multiplication) is not an abelian group. Show that A*B<>B*A for some particular square matrixes with not zero determinant Task 2. (0.2 points) Calculate in not more than 15 steps of multiplication: 15250 mod 13. Show all intermediate steps 15 mod 13 = 2 152 mod 13 =4 154 mod 13 =3 158 mod 13 =9 1516 mod 13 =3 1532 mod 13 =9 1564 mod 13 =3 15128 mod 13 =9 15250 mod 13 = (15128 mod 13)*( 1564 mod 13)*( 1532 mod 13)*( 1516 mod 13)*( 158 mod 13)*( 152 mod 13) = ((9*3) mod 13)*((9*3) mod 13)*9*4) mod 13 = (1*1*9*4) mod 13= 36 mod 13 = 10 Task 3. (0.2 points) Solve equation 6x=5 mod 13 Show all intermediate steps Find 6-1, by Extended Euclid, for example: 11. Then x=6-1*5 mod 13 = 11*5 mod 13 = 55 mod 13 = 3. Task 4. (0.2 points) Find gcd(286, 428) Show all intermediate steps Gcd(286, 428) = gcd(428, 286) = gcd(286, 142) = gcd(142, 2) = gcd(2, 0) = 2 Task 5. (0.2 points) Find 15-1 mod 30 Show all intermediate steps Inverse does not exist because numbers are not relatively prime (30=2*15) Task 6. (0.2 points) Find 15-1 mod 31 Show all intermediate steps Use Extended Euclid: A=(1,0,31), B=(0,1,15) Q=2 T=A-QB=(1,-2,1) Hence, 15-1=-2=29. Actually, 29*15 mod 31=435 mod 31 = 1 Task 7. (0.2 points) Find multiplicative inverse for b(x)= x 7 x 2 1 mod x 8 x 4 x 3 x 1 in GF(28) Show all intermediate steps Apply Extended Euclid: A=(1,0, x 8 x 4 x 3 x 1 ) B=(0,1, x 7 x 2 1 ) Q=x T=A-QB=(1, x, x 4 1 ) A=(0,1, x 7 x 2 1 ) B=(1, x, x 4 1 ) Q= x 3 T=A-QB=( x 3 , x 4 1 , x 3 x 2 1 ) A=(1, x, x 4 1 ) B=( x 3 , x 4 1 , x 3 x 2 1 ) Q=x+1 T=A-QB=( x 4 x 1 , x 5 x 4 1 , x 2 x ) A=( x 3 , x 4 1 , x 3 x 2 1 ) B=( x 4 x 1 , x 5 x 4 1 , x 2 x ) Q=x T=A-QB=( x 5 x 3 x 2 x , x 6 x 5 x 4 x 1 , 1) Hence, inverse is x 6 x 5 x 4 x 1 Task 8. (0.2 points) Compare efforts of brute-force and meet-in-the-middle attacks on Double DES. Efforts of MIM attack on Double DES is 256 on average. Efforts of brute force attack on Double DES is 2111, which is significantly greater. Task 9. (0.2 points) Assume that p=3, q=11. Encrypt and decrypt message M=5 using RSA Show all intermediate steps N=pq=33 Fi=20 E=3 (say) Gcd(e,fi)=1 e-1=d=7 C=Me mod n =53mod 33 = 26 Cd mod n = 267 mod 33 = ((676 mod 33) * (265 mod 33) ) mod 33 = ((16*16) mod 33 *(263 mod 33) ) mod 33 = (25*16*26) mod 33 = (4*26) mod 33 = 104 mod 33 = 5 = M Task 10. (0.2 points) Determine what operations and in what number are necessary to output next byte of generated by RC4 pseudo-random stream (fill a table with two columns: Operation Name, Operation Number). Operation Operation Number + 3 Mod 3 Swap 1 Store 4