The membership problem for unmixed polynomial ideals is solvable in single exponential time
Discrete Applied Mathematics - Special volume on applied algebra, algebraic algorithms, and error-correcting codes
Efficient computation of zero-dimensional Gro¨bner bases by change of ordering
Journal of Symbolic Computation
Ideals, Varieties, and Algorithms: An Introduction to Computational Algebraic Geometry and Commutative Algebra, 3/e (Undergraduate Texts in Mathematics)
Fully homomorphic encryption using ideal lattices
Proceedings of the forty-first annual ACM symposium on Theory of computing
On lattices, learning with errors, random linear codes, and cryptography
Journal of the ACM (JACM)
QUAD: A multivariate stream cipher with provable security
Journal of Symbolic Computation
Fast Cryptographic Primitives and Circular-Secure Encryption Based on Hard Learning Problems
CRYPTO '09 Proceedings of the 29th Annual International Cryptology Conference on Advances in Cryptology
A fully homomorphic encryption scheme
A fully homomorphic encryption scheme
The Learning with Errors Problem (Invited Survey)
CCC '10 Proceedings of the 2010 IEEE 25th Annual Conference on Computational Complexity
Additively homomorphic encryption with d-operand multiplications
CRYPTO'10 Proceedings of the 30th annual conference on Advances in cryptology
Lattice Polly Cracker cryptosystems
Journal of Symbolic Computation
Better key sizes (and attacks) for LWE-based encryption
CT-RSA'11 Proceedings of the 11th international conference on Topics in cryptology: CT-RSA 2011
Homomorphic encryption: from private-key to public-key
TCC'11 Proceedings of the 8th conference on Theory of cryptography
Implementing Gentry's fully-homomorphic encryption scheme
EUROCRYPT'11 Proceedings of the 30th Annual international conference on Theory and applications of cryptographic techniques: advances in cryptology
New algorithms for learning in presence of errors
ICALP'11 Proceedings of the 38th international colloquim conference on Automata, languages and programming - Volume Part I
Probabilistic multivariate cryptography
VIETCRYPT'06 Proceedings of the First international conference on Cryptology in Vietnam
Efficient Fully Homomorphic Encryption from (Standard) LWE
FOCS '11 Proceedings of the 2011 IEEE 52nd Annual Symposium on Foundations of Computer Science
Fully homomorphic encryption with relatively small key and ciphertext sizes
PKC'10 Proceedings of the 13th international conference on Practice and Theory in Public Key Cryptography
Fully homomorphic encryption over the integers
EUROCRYPT'10 Proceedings of the 29th Annual international conference on Theory and Applications of Cryptographic Techniques
ASIACRYPT'11 Proceedings of the 17th international conference on The Theory and Application of Cryptology and Information Security
ASIACRYPT'11 Proceedings of the 17th international conference on The Theory and Application of Cryptology and Information Security
Lattice signatures without trapdoors
EUROCRYPT'12 Proceedings of the 31st Annual international conference on Theory and Applications of Cryptographic Techniques
Polly cracker, revisited, revisited
PKC'12 Proceedings of the 15th international conference on Practice and Theory in Public Key Cryptography
On the complexity of solving quadratic Boolean systems
Journal of Complexity
Solving polynomial systems over finite fields: improved analysis of the hybrid approach
Proceedings of the 37th International Symposium on Symbolic and Algebraic Computation
A fully homomorphic cryptosystem with approximate perfect secrecy
CT-RSA'13 Proceedings of the 13th international conference on Topics in Cryptology
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We initiate the formal treatment of cryptographic constructions ("Polly Cracker") based on the hardness of computing remainders modulo an ideal over multivariate polynomial rings. We start by formalising the relation between the ideal remainder problem and the problem of computing a Gröbner basis. We show both positive and negative results. On the negative side, we define a symmetric Polly Cracker encryption scheme and prove that this scheme only achieves bounded CPA security. Furthermore, we show that a large class of algebraic transformations cannot convert this scheme to a fully secure Polly-Cracker-style scheme. On the positive side, we formalise noisy variants of the ideal membership, ideal remainder, and Gröbner basis problems. These problems can be seen as natural generalisations of the LWE problem and the approximate GCD problem over polynomial rings. We then show that noisy encoding of messages results in a fully IND-CPA-secure somewhat homomorphic encryption scheme. Our results provide a new family of somewhat homomorphic encryption schemes based on new, but natural, hard problems. Our results also imply that Regev's LWE-based public-key encryption scheme is (somewhat) multiplicatively homomorphic for appropriate choices of parameters.