Let ui ∈{0,1} and define f(u0, u1, u2, u3)=(u0+u1, u0, u1+u2, u2+u3) 2 Given an 8 -bit word, let

Let ui ∈{0,1} and define f(u0, u1, u2, u3)=(u0+u1, u0,...

Key Concepts

Binary Operations in GF(2)

This concept involves performing arithmetic on binary digits (bits) where addition is carried out modulo 2. In such a system, the addition operation is equivalent to the XOR (exclusive OR) operation, meaning that 1+1 is 0, and 1+0 or 0+1 gives 1. Bitwise computations using these operations are fundamental in digital logic and cryptography.

Linear Transformations over GF(2)

A linear transformation over GF(2) refers to a function that maps a vector of bits to another vector of bits using operations defined over the finite field GF(2). These transformations are represented by matrices whose entries are in {0,1}, and the function f described is an example of such a transformation. The linearity property is crucial in analyzing and understanding the behavior of cryptographic functions.

Feistel Network Structure

A Feistel network is a structure often used in the design of block ciphers. It divides the input data into two halves, applying a function (the round function) to one half and then combining it with the other half, typically through an XOR operation. The process involves swapping the data halves and iterating the rounds, which helps in achieving diffusion and confusion properties essential for secure encryption.

Iterated (Round) Function Applications

This concept involves successive application of a transformation or round function to process data through multiple iterations. Each iteration modifies the state by applying a well-defined function, often including splitting, transformation, and recombination steps. This iterative process is central to many cryptographic algorithms, where the cumulative effect of multiple rounds produces a secure output from a simple repeated operation.

Transcript

00:01 Hello there.

00:01 Okay, so for this exercise we got this vector, this linear mapping.

00:07 Let's go from r2 to r2 and moreover we know that we can evaluate the evaluation of two vectors.

00:14 Zero one, two and zero one, give us the result to three and one three respectively after applying the map.

00:23 Okay, so we need to find the formula for f.

00:28 So how is defined f? for that, we have defined f.

00:32 For that, we have going to use the theorem 5 .2 and consider that the basis for r2 is defined to the vectors 1 -2 and 01, okay, the ones that we have obtained here.

01:06 So basically this theorem say that given two vector spaces v and u over the field and v has a basis of elements v i from i equals to one up to n and there are vectors ui from i equals to one up to n on u, not necessarily a basis.

01:45 Then there exists a unique map from v to u such that f applied to v is equals to ui for all the i is possible from 1 to n.

02:06 So this is basically what the theorem 522 say, and we are going to use this fact to obtain the formula for the map.

02:17 Okay, so let's take a look to what this theorem say.

02:29 Let's say that we're going to map the element from the basis to to any vector on the image of this mapping.

02:40 So if this is a basis for v, well actually the span of the set is equals to v, then we can define any vector ab on any vector av on r2 as a linear combination of the elements on the basis.

03:15 That means 01 and 12.

03:22 In other words, this is equivalent to say that a, b is equal to x12 plus y 01.

03:37 It's just a linear combination...

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