CertSGN

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We will briefly describe here the contributions of the article[1], in which members of the certSign research group introduced a new method for producing the following:

  • Starting with an encryption scheme which is homomorphic with respect to one operation (such as multiplication), the recipe of the authors produces another encryption scheme which is now homomorphic with respect to two operations (for example, addition and multiplication).

The authors use this technique to construct examples of encryption schemes that, theoretically can handle any algebraic function on encrypted data.

The homomorphic encryption scheme CSGN, a symmetric homomorphic encryption scheme with plaintext (the field with two elements) was introduced in the same article. The latter plays an essential role the architecture of a privacy-preserving contact tracing application, developed by certSign as part of the TAMEC project. [2]

The content of the article is protected under the law by two patents. [3]


Ring homomorphic encryption schemes from monoidal ones. The blueprint

Given a monoid and any commutative ring with unity, one can associate to these an algebra . As an -module, the monoid algebra is free with a basis consisting of the symbols , where .

The multiplication in this algebra is defined by extending in an -bilinear manner. Any element has a unique representation

where for all but finitely many . The product of two elements is given by

We note that the identity element with respect to multiplication is , where is the identity element of . If is a group then the monoid algebra described above is called a group algebra.

If are two monoids, a monoid homomorphism induces a natural -algebra homomorphism .

We also remark that for any -algebra , there is a natural -algebra homomorphism given by

.

The patented blueprint

Let be a finite ring and let be a monoid homomorphic encryption scheme. This implies that the ciphertext and the plaintext are monoids and the decryption algorithm is a monoid homomorphism.

Consider an efficiently computable -character , where is a finite -algebra.

As explained above, the monoid homomorphism induces the -algebra homomorphism . At the same time, induces an -algebra homomorphism . The -algebra homomorphism is defined as

,

defined by the formula

.

Let us denote by the image of in . For the decryption algorithm to remain secure, one needs the assumption that is not the trivial character, a condition that is always assumed in the blueprint.

We now describe a ring homomorphic encryption scheme with plaintext and ciphertext . Let be the image of in and fix some integer such that the elements of the form , with is the whole -algebra .

  • Consider a fixed tuple . For a plaintext consider such that . Then
.
  • The decryption algorithm is given by

The authors of the article prove that is a ring homomorphic encryption scheme and the security of this scheme is the same as the security of the monoid encryption scheme that one starts with.

A symmetric ring homomorphic encryption scheme

  1. M. Barcau and V. Pasol, Bounded Fully Homomorphic Encryption from Monoid Algebras, https://eprint.iacr.org/2018/584
  2. https://www.certsign.ro/en/projects
  3. U.S. Patent Appln. No. 14/936,097 and European Patent Appln. No. EP 15193706.7