Malleable Encryption - Computerphile

Multiplication and Addition: A Similar Connection

In exploring the concept of homomorphic encryption, we often find ourselves dealing with properties that are similar to multiplication and addition. One such example is the connection between these two operations. By raising a cipher text to a power and then multiplying it by another number, we can create an equation where addition is equivalent to multiplication. For instance, if we have a cipher text 'G' and raise it to the power of 42, then multiply it by another number, such as 12, we get G^42 * (G^12). This expression can be rewritten as G^(42+12) = G^54. By recognizing this equivalence, we can use addition instead of multiplication in our encryption schemes.

The Implications of Small Bases

When working with homomorphic encryption, it's essential to consider the base of the exponentiation operation. If the base is very small, such as 54 for a computer, we can calculate the exponents for each possible value and create a table. This allows us to quickly determine the result of any given exponentiation operation without having to compute the logarithm. For example, if we have a cipher text 'G' and want to compute G^54, we can look up the corresponding value in our table instead of performing the actual calculation.

Homomorphic Encryption for Voting Schemes

One of the most significant applications of homomorphic encryption is in secure voting systems. In this context, we need to ensure that individual votes remain confidential while still allowing the tallying of votes to be performed securely. To achieve this, we can use a system where voters encrypt their vote using either G^1 or G^0, depending on whether they voted yes or no. By multiplying the encrypted values together, we can compute the total number of yes votes without revealing individual results. This process can be completed by a central authority who knows the decryption key 'd', which is then used to retrieve the original value of G.

However, using RSA for homomorphic encryption in this context poses an issue. The scheme relies on deterministic encryption, where each encrypted value corresponds to a unique message. Since there are only two possible values (G^0 or G^1), an attacker can quickly learn both values and determine who voted yes or no. This is known as a non-secure voting scheme.

Deterministic Encryption vs. Non-Deterministic Schemes

There exist alternative schemes that offer homomorphic encryption without the drawbacks of deterministic encryption. One such example is Algal, which allows for secure computation without requiring the decryption key to be used in every step of the process. Another notable example is the P crypto system, which simplifies the process even further by making multiplication equivalent to addition.

The P Crypto System: A Simplified Approach

The P crypto system offers a more streamlined approach to homomorphic encryption. By using a different mathematical structure, it eliminates the need for logarithms and makes computation even more efficient. This system is designed to work with addition instead of multiplication, which can simplify the encryption process. However, its security relies on the assumption that an attacker cannot combine their knowledge from two separate attacks.

The Importance of Collaboration

In exploring homomorphic encryption, it's essential to recognize the importance of collaboration between researchers and developers. When working together, they can identify potential vulnerabilities and develop solutions that improve the security and usability of these systems. For instance, combining Algal with RSA might provide a more secure voting system than using RSA alone.

The Future of Homomorphic Encryption

As research in homomorphic encryption continues to evolve, we can expect to see new applications and improvements in existing ones. By pushing the boundaries of what is possible with this technology, we can create more efficient, secure, and accessible systems for various applications. The development of deterministic encryption schemes like RSA has provided a foundation for these advancements, but ongoing research will be necessary to overcome the limitations of current systems.

References:

1. RSA: An overview of RSA encryption

2. Algal: A homomorphic encryption scheme with deterministic encryption

3. P Crypto System: A simplified approach to homomorphic encryption