1.9. Compute x as far as possible without a calculator. Where appropriate, make
ID: 3808285 • Letter: 1
Question
1.9. Compute x as far as possible without a calculator. Where appropriate, make use
of a smart decomposition of the exponent as shown in the example in Sect. 1.4.1:
1. x = 32 mod 13
2. x = 72 mod 13
3. x = 310 mod 13
4. x = 7100 mod 13
5. 7x = 11 mod 13
The last problem is called a discrete logarithm and points to a hard problem which
we discuss in Chap. 8. The security of many public-key schemes is based on the
hardness of solving the discrete logarithm for large numbers, e.g., with more than
1000 bits.
Explanation / Answer
The modulo between two numbers give the remainder of when the first number is divided by the second number.
For example:
a mod b gives the remainder when a is divided by b.
a = q.b + r
a is the dividend, b is the divisior, and r is the remainder.
The result of a mod b can be written as follows:
r = a – q.b … (1)
1.
The 32 mod 13 can be calculated as follows:
On dividing the 32 by 13 the quitionet comes as 2.
Put the value of q = 2 in equation (1).
x = 32 – 2 * 13
= 32 – 26
= 6
Hence, the value of x is 6.
2.
The 72 mod 13 can be calculated as follows:
On dividing the 72 by 13 the quitionet comes as 5.
Put the value of q = 5 in equation (1).
x = 72 – 5 * 13
= 72 – 65
= 7
Hence, the value of x is 7.
3.
The 310 mod 13 can be calculated as follows:
On dividing the 310 by 13 the quitionet comes as 23.
Put the value of q = 23 in equation (1).
x = 310 – 23 * 13
= 310 – 299
= 11
Hence, the value of x is 11.
4.
The 7100 mod 13 can be calculated as follows:
On dividing the 7100 by 13 the quitionet comes as 546.
Put the value of q = 546 in equation (1).
x = 7100 – 546 * 13
= 7100 – 7098
= 2
Hence, the value of x is 2.
5.
The 11 mod 13 can be calculated as follows:
On dividing the 11 by 13 the quitionet comes as 0.
Put the value of q = 0 in equation (1).
11 – 0 * 13
= 11 – 0
= 11
7x = 11
Take log at both sides.
log 7x = log 11 … (2)
Property 1: log ab = b log a
Using property 1, the expression (2) can be simplified as dollows:
x log 7 = log 11
x = log 11/log 7 … (3)
Property 2: log a/log b = log (a – b)
Using property 2, the expression (3) can be simplified as follows:
x = log (11 – 7)
x = log (4) … (4)
Using property 1, the expression (4) can be simplified as follows:
x = log (22) … (5)
Using property 1, the expression (5) can be simplified as follows:
x = 2 log (2)
Since the value of the log (2) = 0.301. Thus, put the value of the log (2) in the x = 2 log (2).
x = 2 * 0.301
x = 0.602
Hence, the value of x is 0.60.
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