COMP 547: Assignment 1 Solutions

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+ x^127 + x^124 + x^122 + x^121 + x^119 + x^117 + x^114 + x^113 + x^109 + x^108 + x^106 + x^105 + x^104 + x^103 + x^101 + x^94 + x^92 + x^89 + x^86 + x^83 + x^77 + x^75 + x^72 + x^71 + x^70 + x^69 + x^67 + x^65 + x^62 + x^60 + x^59 + x^58 + x^56 + x^53 + x^51 + x^48 + x^46 + x^44 + x^43 + x^42 + x^41 + x^40 + x^39 + x^37 + x^36 + x^35 + x^34 + x^32 + x^30 + x^27 + x^25 + x^24 + x^20 + x^17 + x^16 + x^11 + x^10 + x^9 + x^7 + x^6 + x^4 + x^3 + x; 5. The x, y pair used above, such that g x = i and g y = j, was as follows: x := 9996624267323432385860154688013965289244775912541095846\ 535852465958036219324187129020612749492978345054637631822\ 068685180242273194079032073433587702733320902621411219030\ 197361870983920382212856316640364750228006953961590545028\ 536588900708607519411648639259588779023984224837585211337\ 596610575030089733; y := 4421400686325875904669189606687247157171896340333697491\ 038345589908751327223429043940799468738274631257310849856\ 475555535414243218803857633670804928823838373666740772907\ 628430736972473471959749800914977273140890859864815145939\ 651743208870161801887951726958872861225054888743539321784\ 292172995856175649; 6. Here is a randomly generated 1000-bit message string m: m := "011101101101010010000100110001000001000011011101100000\ 011100111100010110111001110101001011001101001101001100111\ 001010010110000010011111110101001110111000000111001110010\ 110110001100101101101110100111010100001000111001010010110\ 011010100010111101010110000101011000111011011100101011000\ 001000110001100010011001111100000101100111111110000001010\ 101011110001000001010111111001011100011001000010100011110\ 111100011100001001100010100111010011000100111111011010000\ 110001111010100011000101110101001011101111000001110011011\ 010011101110101101010011010111100111110101000001000010001\ 011001100110100000011001100100100001000100101001000111100\ 011010011101011000111010110110011110111011101010101111111\ 010001011001111010001101010111100010010101010010001010001\ 011011111100010010010110101110101100110011010111000011000\ 001001110010111110111001010111011001011111000001000000011\ 001000000100001000001011011011010111111011000001100010000\ 001110010101100001101110001010001000010100101100110100001\ 1011011001101001011001010000001110"; 10
The following Maple function computes the tag of this message from i and j : tag := proc(Field, m, i, j) local msg, fulltag, len; msg := Field[input](convert(m, decimal, binary)); fulltag := convert(convert(Field[output](Field[‘+‘]( \ Field[‘*‘](msg, i), j)), binary), string); len := length(fulltag); return(substring(fulltag, len-49..len)); end proc; This generated the following tag t: t := "01110011001001111001011000000110100110100100101000"; 7. Here is the function pqsrt: pqsqrt := proc(x, p, q) local r0, r1; if not (isprime(p) and isprime(q)) then return(’FAIL’); end if; r0 := msqrt(x,p); r1 := msqrt(x,q); if r0 = ’FAIL’ or r1 = ’FAIL’ then return(’FAIL’); end if; if p = q then return msqrt(x, p^2); end if; return(chrem([r0, r1], [p, q])); end proc; 8. The square root modulo n problem, where n is composite, is a computationally difficult problem, equivalent to factoring n. Thus, we will not spend much effort on attempting to solve it in general. However, we have efficient algorithms for solving the square root modulo p problem, where p is prime. With these algorithms and the Chinese remainder theorem, we are able to solve the square root modulo n problem, where n is a composite integer whose prime factorisation n = pq is known. Maple’s msqrt function is efficient if the given modulus is prime (or in certain other special cases, such as if the given modulus is a square of a prime). However, given a nonprime modulus n (even if we, the user, know its factorisation), msqrt is unable to efficiently search for the requested 11
Page 1 and 2: COMP 547: Assignment 1 Solutions Oc
Page 3 and 4: + x^277 + x^276 + x^275 + x^272 + x
Page 5 and 6: g := x^997 + x^996 + x^994 + x^991
Page 7 and 8: We know that the distinct powers of
Page 9: + x^790 + x^788 + x^784 + x^783 + x
Page 13 and 14: a:= 0; b:= 0; while not (a = -1 and
Page 15 and 16: 21284645683965405414161580243440656
Page 17 and 18: 17819930918348761695484963068116150
Page 19 and 20: 38925360540593985506091325701956938
Page 21 and 22: 5471234567945 is in QR_n (root of x
Page 23 and 24: 5471234567962 is not in QR_n (root
Page 25 and 26: 5471234567979 is not in QR_n (root
Page 27 and 28: integers in X fall into either of t
Page 29 and 30: There are simply too many condition
Page 31 and 32: (a + √ x) p−1 2 (a − √ x) p
Page 33 and 34: Thus, if rootLV fails, then a does
Page 35 and 36: Now, armed with these lemmas, let u

COMP 547: Assignment 1 Solutions

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