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# GATE | Gate IT 2007 | Question 15

Consider the following two statements:
i. A hash function (these are often used for computing digital signatures) is an injective function.
A. encryption technique such as DES performs a permutation on the elements of its input alphabet.
Which one of the following options is valid for the above two statements?
(A) Both are false
(B) Statement (i) is true and the other is false
(C) Statement (ii) is true and the other is false
(D) Both are true

Explanation:

Injective Function:

A function F(X) is said to be injective if it has one-to-one mapping.
Statement 1: Hash function is an injective function
Statement 2: DES Encryption technique performs a permutation on the elements of its input alphabet

1) Generally, a hash function H(X) is mapping from a larger set to a predefined output set
For example, let H(X) = (X)%5
The above function H(X) is not injective because
Let X1 = 10, X2 = 15
H(10) = H(15) = 0
As the output of H(X1) = H(X2) where X1!= X2 => H(X) is many-to-one function.
Statement 1 is false.

2) In DES encryption scheme, it performs P-Box permutation.
Statement 1 is false,
Statement 2 is true.

This solution is contributed by Anil Saikrishna Devarasetty.

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