Question

Explain why, if you were encoding messages \(x_{1}, x_{2}\), and \(x_{3}\) to obtain \(y_{1}, y_{2}\), and \(y_{3}\) by adding an arbitrary number \(a\) and taking the sum \(\bmod n\), your adversary would know that at least one of the differences \(y_{1}-y_{2}, y_{1}-y_{3}\), or \(y_{2}-y_{3}\) taken in the integers, not in \(Z_{n}\), would be the difference of two unencoded messages. (Note: We are not saying that your adversary would know which of the three was such a difference.)

Short answer

An adversary knows one computed difference matches an original difference.

Step-by-step solution

Understanding the Message Encoding

Consider three messages: \(x_1, x_2,\) and \(x_3\). These messages are encoded to \(y_1, y_2,\) and \(y_3\) by adding an arbitrary number \(a\) to each message, so \(y_i = (x_i + a) \mod n\) for \(i = 1, 2, 3\). The operation of adding \(a\) and taking modulo \(n\) does not change the differences between the messages themselves.

Defining Message Differences

The differences between these encoded messages, taken as integers, are: \(y_1 - y_2 = (x_1 + a) - (x_2 + a)\), \(y_1 - y_3 = (x_1 + a) - (x_3 + a)\), and \(y_2 - y_3 = (x_2 + a) - (x_3 + a)\). Simplifying these, we find they are equivalent to \(x_1 - x_2\), \(x_1 - x_3\), and \(x_2 - x_3\) respectively, regardless of \(a\) or \(n\).

Analyzing Differences

Since the arbitrary constant \(a\) and the modulo operation do not affect the difference, the differences \(y_1 - y_2\), \(y_1 - y_3\), and \(y_2 - y_3\) in integers are exactly \(x_1 - x_2\), \(x_1 - x_3\), and \(x_2 - x_3\). An adversary could calculate these differences from the encoded messages \(y_1, y_2,\) and \(y_3\).

Concluding the Adversary's Knowledge

Even though the adversary cannot know which specific difference corresponds to which pair of original messages, they can be certain that at least one of the computed differences between the encoded messages corresponds directly to a difference between two of the original messages \(x_1, x_2,\) or \(x_3\).

Key concepts

Message Encoding

In message encoding, the goal is to transform original messages into encoded versions, often by using a specific mathematical operation. The exercise here involves encoding messages by adding a constant number to each message and then applying the modulo operation.
These steps mean turning a message like \( x_1 \) into its encoded counterpart \( y_1 \) by calculating \( y_1 = (x_1 + a) \mod n \). Here, \( a \) is an arbitrary number chosen to encode the message, and \( n \) is the modulus used in the encoding process.
This process is intended to hide the original messages by transforming them into a seemingly unrelated set of numbers. However, subtle properties of the modulo operation may still reveal some underlying relationships between the original messages, which is critical in ensuring both security and secrecy of the communication.

Modulo Operation

The modulo operation is a fundamental concept in mathematics, and it allows you to calculate the remainder of a division between two numbers. In this exercise, it's used in the encoding process to transform messages. When an original message \( x_i \) is added to a constant \( a \), the sum is then computed modulo \( n \), resulting in \( y_i = (x_i + a) \mod n \).
This operation wraps around the number line, restricting the encoded message to a defined range from 0 to \( n-1 \). While this helps mask the original value, the operation has a unique feature—it does not change differences between the values it processes.
Through this wrapping effect, the modulo operation allows certain patterns to remain detectable, even if the exact values are obscured. This characteristic is crucial when analyzing encoded messages, as it maintains the differences between messages even after encoding.

Integer Differences

Understanding integer differences is vital because even after encoding messages, differences between original values remain unchanged.
Let's consider three original messages \( x_1, x_2, \) and \( x_3 \) that are encoded to \( y_1, y_2, \) and \( y_3 \) respectively. Each of these encoded values represents an equation of the form \( y_i = (x_i + a) \mod n \).
The exercise highlights an important feature: despite the encoding, the integer differences are preserved. This means that the encoded differences \( y_1-y_2 \), \( y_1-y_3 \), and \( y_2-y_3 \) essentially reflect the unencoded differences \( x_1-x_2 \), \( x_1-x_3 \), and \( x_2-x_3 \).

• This unanticipated side effect means an observer can infer at least one of these original differences from the encoded messages.
• By analyzing the differences, we understand why message encoding, despite its transformation, still retains some original information patterns.

This concept demonstrates why understanding integer differences is key to message encoding security, as it reveals how certain properties of numbers survive transformations and can impact secrecy.