In this example, we wish to add 1 and 2. The data is encrypted so that 1 becomes 33 and 2 becomes The encrypted data is sent to the cloud and processed: the result 87 can be downloaded from the cloud and decrypted to provide the final answer 3. Now Gentry, an IBM researcher, has shown that it is possible to analyze data without decrypting it. The key is to encrypt the data in such a way that performing a mathematical operation on the encrypted information and then decrypting the result produces the same answer as performing an analogous operation on the unencrypted data. The correspondence between the operations on unencrypted data and the operations to be performed on encrypted data is known as a homomorphism.

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Description[ edit ] Homomorphic encryption is a form of encryption with an additional evaluation capability for computing over encrypted data without access to the secret key. The result of such a computation remains encrypted.

Homomorphic encryption can be viewed as an extension of either symmetric-key or public-key cryptography. Homomorphic refers to homomorphism in algebra: the encryption and decryption functions can be thought of as homomorphisms between plaintext and ciphertext spaces.

Homomorphic encryption includes multiple types of encryption schemes that can perform different classes of computations over encrypted data. The computations are represented as either Boolean or arithmetic circuits. Partially homomorphic encryption encompasses schemes that support the evaluation of circuits consisting of only one type of gate, e.

Somewhat homomorphic encryption schemes can evaluate two types of gates, but only for a subset of circuits. Leveled fully homomorphic encryption supports the evaluation of arbitrary circuits of bounded pre-determined depth. Fully homomorphic encryption FHE allows the evaluation of arbitrary circuits of unbounded depth, and is the strongest notion of homomorphic encryption.

For the majority of homomorphic encryption schemes, the multiplicative depth of circuits is the main practical limitation in performing computations over encrypted data. Homomorphic encryption schemes are inherently malleable. In terms of malleability, homomorphic encryption schemes have weaker security properties than non-homomorphic schemes. History[ edit ] Homomorphic encryption schemes have been developed using different approaches.

Specifically, fully homomorphic encryption schemes are often grouped into generations corresponding to the underlying approach. During that period, partial results included the following schemes: RSA cryptosystem unbounded number of modular multiplications ; ElGamal cryptosystem unbounded number of modular multiplications ; Goldwasser—Micali cryptosystem unbounded number of exclusive or operations ; Benaloh cryptosystem unbounded number of modular additions ; Paillier cryptosystem unbounded number of modular additions ; Sander-Young-Yung system after more than 20 years solved the problem for logarithmic depth circuits ; [4] Boneh—Goh—Nissim cryptosystem unlimited number of addition operations but at most one multiplication ; [5] First-generation FHE[ edit ] Craig Gentry , using lattice-based cryptography , described the first plausible construction for a fully homomorphic encryption scheme.

The construction starts from a somewhat homomorphic encryption scheme, which is limited to evaluating low-degree polynomials over encrypted data; it is limited because each ciphertext is noisy in some sense, and this noise grows as one adds and multiplies ciphertexts, until ultimately the noise makes the resulting ciphertext indecipherable. Gentry then shows how to slightly modify this scheme to make it bootstrappable, i.

Finally, he shows that any bootstrappable somewhat homomorphic encryption scheme can be converted into a fully homomorphic encryption through a recursive self-embedding. By "refreshing" the ciphertext periodically whenever the noise grows too large, it is possible to compute an arbitrary number of additions and multiplications without increasing the noise too much. Gentry based the security of his scheme on the assumed hardness of two problems: certain worst-case problems over ideal lattices , and the sparse or low-weight subset sum problem.

The somewhat homomorphic component in the work of Van Dijk et al. The Levieil—Naccache scheme supports only additions, but it can be modified to also support a small number of multiplications.

Many refinements and optimizations of the scheme of Van Dijk et al. Second-generation FHE[ edit ] The homomorphic cryptosystems in current use are derived from techniques that were developed starting in by Zvika Brakerski, Craig Gentry , Vinod Vaikuntanathan, and others. These innovations led to the development of much more efficient somewhat and fully homomorphic cryptosystems. This NTRU variant was subsequently shown vulnerable to subfield lattice attacks, [25] [24] which is why these two schemes are no longer used in practice.

A distinguishing characteristic of the second-generation cryptosystems is that they all feature a much slower growth of the noise during the homomorphic computations. Additional optimizations by Craig Gentry , Shai Halevi , and Nigel Smart resulted in cryptosystems with nearly optimal asymptotic complexity: Performing T operations on data encrypted with security parameter k.

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## What Is Homomorphic Encryption? And Why Is It So Transformative?

Just like other forms of encryption, homomorphic encryption uses a public key to encrypt the data. Then, only the individual with the matching private key can access the unencrypted data after the functions and manipulation are complete. This allows the data to be and remain secure and private even when someone is using it. There are three main types of homomorphic encryption: partially homomorphic encryption keeps sensitive data secure by only allowing select mathematical functions to be performed on encrypted data ; somewhat homomorphic encryption supports limited operations that can be performed only a set number of times ; fully homomorphic encryption this is the gold standard of homomorphic encryption that keeps information secure and accessible. They can only take the raw materials and create something inside the box. When they finish, the person who has the key can remove the materials processed data.

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## Craig Gentry (computer scientist)

Description[ edit ] Homomorphic encryption is a form of encryption with an additional evaluation capability for computing over encrypted data without access to the secret key. The result of such a computation remains encrypted. Homomorphic encryption can be viewed as an extension of either symmetric-key or public-key cryptography. Homomorphic refers to homomorphism in algebra: the encryption and decryption functions can be thought of as homomorphisms between plaintext and ciphertext spaces. Homomorphic encryption includes multiple types of encryption schemes that can perform different classes of computations over encrypted data. The computations are represented as either Boolean or arithmetic circuits. Partially homomorphic encryption encompasses schemes that support the evaluation of circuits consisting of only one type of gate, e.

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## Hacker Lexicon: What Is Homomorphic Encryption?

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