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Questions and Answers

Which of the following statements is true about the negative of a point P on an elliptic curve?

  • The x-coordinate of -P is negative, but the y-coordinate is the same as that of P.
  • The x-coordinate of -P is positive, but the y-coordinate is the same as that of P.
  • The x-coordinate of -P is the same as that of P, but the y-coordinate is positive.
  • The x-coordinate of -P is the same as that of P, but the y-coordinate is negative. (correct)

If P and Q are two points on an elliptic curve with different x coordinates, how do we define P + Q?

  • P + Q is the mirror image of the third point of intersection of the straight line drawn between P and Q. (correct)
  • P + Q is the third point of intersection of the straight line drawn between P and Q.
  • P + Q is the negative of the third point of intersection of the straight line drawn between P and Q.
  • P + Q is the sum of the x coordinates of P and Q.

What is the result of adding a point P to its negative -P on an elliptic curve?

  • The result is the negative of the y-coordinate of P.
  • The result is the negative of the x-coordinate of P.
  • The result is the point at infinity O. (correct)
  • The result is the point P.

What happens when a straight line is tangent to the curve at either point P or Q?

<p>The third point of intersection R is taken as P or Q, respectively. (B)</p> Signup and view all the answers

What is the geometric interpretation of two points P and -P with the same x coordinate on an elliptic curve?

<p>The points are joined by a vertical line that intersects the curve at the infinity point. (B)</p> Signup and view all the answers

What is the result of adding the point at infinity O to any point P on an elliptic curve?

<p>The result is the point P. (D)</p> Signup and view all the answers

What is the result of adding two points P and Q with the same x coordinate on an elliptic curve?

<p>The result is the point at infinity O. (D)</p> Signup and view all the answers

What is the result of adding the point at infinity O to the negative of a point P on an elliptic curve?

<p>The result is the negative of the x-coordinate of P. (A)</p> Signup and view all the answers

What is the result of adding the point at infinity O to itself on an elliptic curve?

<p>The result is the point at infinity O. (B)</p> Signup and view all the answers

What is the result of adding a point P to the point at infinity O on an elliptic curve?

<p>The result is the point P. (D)</p> Signup and view all the answers

Which of the following is a key component in the Elgamal cryptosystem?

<p>Public key (C)</p> Signup and view all the answers

What is the purpose of using a one-time key in the Elgamal cryptosystem?

<p>To encrypt the plaintext message (B)</p> Signup and view all the answers

What is the security of the Elgamal cryptosystem based on?

<p>Computing discrete logarithms (A)</p> Signup and view all the answers

According to the text, what could make the new protocol vulnerable to an attack?

<p>The simplification proposed by the girl's father (C)</p> Signup and view all the answers

According to the text, what does an adversary need to be able to obtain a message sent from user A to user B?

<p>All of the above (D)</p> Signup and view all the answers

According to the text, what does Bob do to obtain the original message M from the intercepted ciphertext C?

<p>All of the above (D)</p> Signup and view all the answers

According to the text, what is the purpose of the Diffie-Hellman key exchange algorithm?

<p>To securely exchange a key for subsequent symmetric encryption of messages (A)</p> Signup and view all the answers

What are the publicly known numbers in the Diffie-Hellman key exchange algorithm?

<p>A prime number q and an integer a (A)</p> Signup and view all the answers

How does user A compute YA in the Diffie-Hellman key exchange algorithm?

<p>YA = aXA mod q (D)</p> Signup and view all the answers

Which of the following is a vulnerability of the key exchange protocol described in the text?

<p>Man-in-the-Middle Attack (A)</p> Signup and view all the answers

What are the global elements of the Elgamal cryptographic system?

<p>Prime number q and primitive root a (B)</p> Signup and view all the answers

How does user A generate a private/public key pair in the Elgamal cryptographic system?

<p>Choose a random integer XA (B)</p> Signup and view all the answers

Which of the following is true about the Diffie-Hellman key exchange?

<p>The security of the key exchange lies in the difficulty of calculating exponentials modulo a prime (C)</p> Signup and view all the answers

What information does an adversary have to work with in order to determine the secret key K in the Diffie-Hellman key exchange?

<p>q, a, YA, YB (A)</p> Signup and view all the answers

In the example given, what is the common secret key K computed by Alice and Bob in the Diffie-Hellman key exchange?

<p>160 (C)</p> Signup and view all the answers

Which equation represents a cubic equation for an elliptic curve over real numbers?

<p>$y^2 = x^3 + ax + b$ (B)</p> Signup and view all the answers

What is the condition that must be met in order to define a group based on the set $E(a, b)$ for specific values of $a$ and $b$ in the equation $y^2 = x^3 + ax + b$?

<p>$a^3 + 27b^2 eq 0$ (D)</p> Signup and view all the answers

What is the geometric interpretation of adding three points on an elliptic curve that lie on a straight line?

<p>The sum of the three points is the point at infinity O. (C)</p> Signup and view all the answers

Which of the following statements is true about the addition of the point at infinity O to any point P on an elliptic curve?

<p>The result is always the point P (C)</p> Signup and view all the answers

What is the geometric interpretation of two points P and -P with the same x coordinate on an elliptic curve?

<p>They are joined by a vertical line that intersects the curve at the infinity point (A)</p> Signup and view all the answers

What is the result of adding a point P to its negative -P on an elliptic curve?

<p>It is always the point at infinity O (A)</p> Signup and view all the answers

What is the result of adding the point at infinity O to the negative of a point P on an elliptic curve?

<p>It is always the point (x, -y) (C)</p> Signup and view all the answers

What is the result of adding a point P to the point at infinity O on an elliptic curve?

<p>It is always the point P (A)</p> Signup and view all the answers

What happens when a straight line is tangent to the curve at either point P or Q?

<p>The line intersects the curve at the point at infinity O (D)</p> Signup and view all the answers

If P and Q are two points on an elliptic curve with different x coordinates, how do we define P + Q?

<p>P + Q is the point of intersection of the line joining P and Q with the curve (A)</p> Signup and view all the answers

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