Let p 23434549 we know that 4842 is larger than radicp

Question

1. Let p = 23434549. We know that 4842 is larger than √P. Suppose it is given that any number less than 4842 does not divide p. What do you think about the primality of p?

2. Let p = 4k + 3 for some integer k and x be a non-zero integer less than p such that it has a square root. Show that square of

x^P+1/4 

is x mod p.

3. Let

E : y² = x³ + 3x + 5 mod 443

be an elliptic curve. Its order is 427 = 61.7.

The point P = (x, y) is on the curve and it is given that 2P = (248,113).

(a) Explain an efficient method to find P = (x, y) (Do not find P, just explain how to find it).

(b) By using P, explain a method to find an element of order 61.