show that for any interger a, a^37 ≡ a (mod 1729) hint: 1729 = 7x13x19
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show that for any interger a,
a^37 ≡ a (mod 1729)
hint: 1729 = 7x13x19
List all points (x,y) in the elliptic curve y2≡ x3 + 2x - 9 (mod 19). (Hint: Corresponding to any...
List all points (x,y) in the elliptic curve y2≡ x3 + 2x - 9 (mod
19). (Hint: Corresponding to any given x , points (x,y) and (x,-y)
can exist on the elliptic curve only if y2≡ x3 + 2x - 9 (mod 19) is
a quadratic residue mod 19. Recall that a value v
∊ Zp is a quadratic residue modulo p only if v(p-1)/2≡ 1 (mod p).
If v is indeed a quadratic residue, we can calculate the two...
4. Show that the following congruence is true ab = (a mod n)" (mod n) for...
4. Show that the following congruence is true ab = (a mod n)" (mod n) for any positive integers a, b, and n.
1. Show that a 1728 = 1 (mod p) when p= 7, 13, 19 for all...
1. Show that a 1728 = 1 (mod p) when p= 7, 13, 19 for all a E N such that p /a. 2. Let p be a prime and p = 3 (mod 4). Show that r2 = -1 (mod p) has no solution. (Hint: Raise both side to (p-1)/2.)
Suppose a c mod n and bd mod n. (a) show that a + b c...
Suppose a
c mod n and bd
mod n.
(a) show that a + b
c + d mod n
(b) show that a * b
c * d mod n.
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1. Show that the number of solutions (x mod p, y mod p) to the equation...
1. Show that the number of solutions (x mod p, y mod p) to the equation x² + 1 = y2 mod p is p- p (6+1) k=0
Let m be a positive integer. Show that a mod m - b mod m t...
Let m be a positive integer. Show that a mod m - b mod m t a - b (mod m) Drag the necessary statements and drop them into the appropriate blank to build your proof (mod m Dag the mecesary eemnes a ohem int the approprite Proof method: Proof assumptions), at-qm + Proof by contradiction aaandh mam it Implication(s) and deduction(s) resulting from the assumption(s): a mk + bmk Hqm tr a-(k + q)m+ r Conclusion(s) from implications and...
For the following LCG generator: Xi = (13Xi1 + 13)(mod(16)), X0 = 37. What is the...
For the following LCG generator: Xi = (13Xi1 + 13)(mod(16)), X0 = 37. What is the maximum possible period length for this generator? Does this generator achieve the maximum possible period length? Justify your answer. Generate 2 pseudo-random uniform numbers for this generator. Xo=37, a=13, c=13, m=16
Let p be an odd prime and a an integer with p not dividing a. Show that a(p-1)/2 is congruent to 1 mod p if and only if a is a square modulo p and -1 otherwise. (hint: think generators)
Let p be an odd prime and a an integer with p not dividing a. Show that a(p-1)/2 is congruent to 1 mod p if and only if a is a square modulo p and -1 otherwise. (hint: think generators)
25. Show that the number of reduced residues a (mod m) such that am-1 1 (mod...
25. Show that the number of reduced residues a (mod m) such that am-1 1 (mod m) is exactly II(p-1,m 1) plm
Show all the steps: 42^7 mod 253 = 15 (5*269) mod 336 =27
Show all the steps: 42^7 mod 253 = 15 (5*269) mod 336 =27
List all points (x,y) in the elliptic curve y2≡ x3 + 2x - 9 (mod
19). (Hint: Corresponding to any given x , points (x,y) and (x,-y)
can exist on the elliptic curve only if y2≡ x3 + 2x - 9 (mod 19) is
a quadratic residue mod 19. Recall that a value v
∊ Zp is a quadratic residue modulo p only if v(p-1)/2≡ 1 (mod p).
If v is indeed a quadratic residue, we can calculate the two...
4. Show that the following congruence is true ab = (a mod n)" (mod n) for any positive integers a, b, and n.
1. Show that a 1728 = 1 (mod p) when p= 7, 13, 19 for all a E N such that p /a. 2. Let p be a prime and p = 3 (mod 4). Show that r2 = -1 (mod p) has no solution. (Hint: Raise both side to (p-1)/2.)
Suppose a
c mod n and bd
mod n.
(a) show that a + b
c + d mod n
(b) show that a * b
c * d mod n.
We were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this imageWe were unable to transcribe this image
1. Show that the number of solutions (x mod p, y mod p) to the equation x² + 1 = y2 mod p is p- p (6+1) k=0
Let m be a positive integer. Show that a mod m - b mod m t a - b (mod m) Drag the necessary statements and drop them into the appropriate blank to build your proof (mod m Dag the mecesary eemnes a ohem int the approprite Proof method: Proof assumptions), at-qm + Proof by contradiction aaandh mam it Implication(s) and deduction(s) resulting from the assumption(s): a mk + bmk Hqm tr a-(k + q)m+ r Conclusion(s) from implications and...
25. Show that the number of reduced residues a (mod m) such that am-1 1 (mod m) is exactly II(p-1,m 1) plm
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