## MA-410 Homework 3

Due at 4:59pm in my mailbox in SAS 3151,
Thursday, April 11, 2019

Solutions may only be submitted in hard copy.
Note my office hours on my
schedule.

- ENT, §6.2, Problem 1, page 116.
- ENT, §8.4, Problem 2, page 167.
Please use the residue 2 for the primitive root modulo 11.
- ENT, §9.1, Problem 8, page 174.
Please prove part (a) for r being a quadratic non-residue.
For part (b) use r=3 as a quadratic non-residue modulo
both 17 and 41.
- ENT, §10.1, Problem 14, page 209.
[Hint: use Maple's “&^ mod” procedure.
There may be a typo in the hint: 1013 ⋅ 17 ≡ 1 (mod φ(2573)).]
*
Note: the 7th edition of the text book has different numbers than the 6th,
which decrypt to actual text:
(n,k) = (2573, 1013) and the cipher text is
0464 1472 0636 1262 2111.
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- Bonus problem:
Let p be a prime ≡ 5 (mod 8);
then p-1 ≡ 0 (mod 4) and p+3 ≡ 0 ≡ 3p+1 (mod 8).

Let a be a quadratic residue, and r a quadratic non-residue,
and let b = 2^{-1}
((1+r^{(p-1)/4}) a^{(3p+1)/8}
+ (1-r^{(p-1)/4}) a^{(p+3)/8}
) mod p. Please prove that b^{2} ≡ a (mod p).