Quiz

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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?

The third point of intersection R is taken as P or Q, respectively. 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?

The points are joined by a vertical line that intersects the curve at the infinity point. 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?

The result is the point 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?

The result is the point at infinity O. 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?

The result is the negative of the x-coordinate of P. Signup and view all the answers

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

The result is the point at infinity O. 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?

The result is the point P. Signup and view all the answers

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

Public key Signup and view all the answers

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

To encrypt the plaintext message Signup and view all the answers

What is the security of the Elgamal cryptosystem based on?

Computing discrete logarithms Signup and view all the answers

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

The simplification proposed by the girl's father 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?

All of the above 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?

All of the above Signup and view all the answers

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

To securely exchange a key for subsequent symmetric encryption of messages Signup and view all the answers

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

A prime number q and an integer a Signup and view all the answers

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

YA = aXA mod q Signup and view all the answers

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

Man-in-the-Middle Attack Signup and view all the answers

What are the global elements of the Elgamal cryptographic system?

Prime number q and primitive root a Signup and view all the answers

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

Choose a random integer XA Signup and view all the answers

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

The security of the key exchange lies in the difficulty of calculating exponentials modulo a prime 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?

q, a, YA, YB 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?

160 Signup and view all the answers

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

$y^2 = x^3 + ax + b$ 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 $y^2 = x^3 + ax + b$?

$a^3 + 27b^2 eq 0$ 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?

The sum of the three points is the point at infinity O. 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?

The result is always the point 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?

They are joined by a vertical line that intersects the curve at the infinity point Signup and view all the answers

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

It is always the point at infinity O 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?

It is always the point (x, -y) 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?

It is always the point P Signup and view all the answers

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

The line intersects the curve at the point at infinity O 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 + Q is the point of intersection of the line joining P and Q with the curve Signup and view all the answers

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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?

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

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

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

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

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

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

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

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

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

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

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

What is the security of the Elgamal cryptosystem based on?

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

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

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

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

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

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

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

What are the global elements of the Elgamal cryptographic system?

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

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

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

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

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

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 $y^2 = x^3 + ax + b$?

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

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

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

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

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

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

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

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

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