Solving sparse linear equations over finite fields
IEEE Transactions on Information Theory
Direct methods for sparse matrices
Direct methods for sparse matrices
Numerical analysis: 4th ed
Public quadratic polynomial-tuples for efficient signature-verification and message-encryption
Lecture Notes in Computer Science on Advances in Cryptology-EUROCRYPT'88
Equations for the projective closure and effective Nullstellensatz
Discrete Applied Mathematics - Special volume on applied algebra, algebraic algorithms, and error-correcting codes
On randomized Lanczos algorithms
ISSAC '97 Proceedings of the 1997 international symposium on Symbolic and algebraic computation
The Design of Rijndael
Computers and Intractability: A Guide to the Theory of NP-Completeness
Computers and Intractability: A Guide to the Theory of NP-Completeness
Some New Effectivity Bounds in Computational Geometry
AAECC-6 Proceedings of the 6th International Conference, on Applied Algebra, Algebraic Algorithms and Error-Correcting Codes
QUARTZ, 128-Bit Long Digital Signatures
CT-RSA 2001 Proceedings of the 2001 Conference on Topics in Cryptology: The Cryptographer's Track at RSA
The Security of Hidden Field Equations (HFE)
CT-RSA 2001 Proceedings of the 2001 Conference on Topics in Cryptology: The Cryptographer's Track at RSA
FLASH, a Fast Multivariate Signature Algorithm
CT-RSA 2001 Proceedings of the 2001 Conference on Topics in Cryptology: The Cryptographer's Track at RSA
Gröbner-Bases, Gaussian elimination and resolution of systems of algebraic equations
EUROCAL '83 Proceedings of the European Computer Algebra Conference on Computer Algebra
Cryptanalysis of the HFE Public Key Cryptosystem by Relinearization
CRYPTO '99 Proceedings of the 19th Annual International Cryptology Conference on Advances in Cryptology
Essential Algebraic Structure within the AES
CRYPTO '02 Proceedings of the 22nd Annual International Cryptology Conference on Advances in Cryptology
Solving Large Sparse Linear Systems over Finite Fields
CRYPTO '90 Proceedings of the 10th Annual International Cryptology Conference on Advances in Cryptology
C*-+ and HM: Variations Around Two Schemes of T. Matsumoto and H. Imai
ASIACRYPT '98 Proceedings of the International Conference on the Theory and Applications of Cryptology and Information Security: Advances in Cryptology
Cryptanalysis of Block Ciphers with Overdefined Systems of Equations
ASIACRYPT '02 Proceedings of the 8th International Conference on the Theory and Application of Cryptology and Information Security: Advances in Cryptology
A Fast and Secure Implementation of Sflash
PKC '03 Proceedings of the 6th International Workshop on Theory and Practice in Public Key Cryptography: Public Key Cryptography
A new efficient algorithm for computing Gröbner bases without reduction to zero (F5)
Proceedings of the 2002 international symposium on Symbolic and algebraic computation
EUROCRYPT'96 Proceedings of the 15th annual international conference on Theory and application of cryptographic techniques
Efficient algorithms for solving overdefined systems of multivariate polynomial equations
EUROCRYPT'00 Proceedings of the 19th international conference on Theory and application of cryptographic techniques
Higher order correlation attacks, XL algorithm and cryptanalysis of Toyocrypt
ICISC'02 Proceedings of the 5th international conference on Information security and cryptology
About the XL algorithm over GF(2)
CT-RSA'03 Proceedings of the 2003 RSA conference on The cryptographers' track
Enumerative Combinatorics: Volume 1
Enumerative Combinatorics: Volume 1
Multivariates Polynomials for Hashing
Information Security and Cryptology
Kipnis-Shamir Attack on HFE Revisited
Information Security and Cryptology
On the Number of Linearly Independent Equations Generated by XL
SETA '08 Proceedings of the 5th international conference on Sequences and Their Applications
Practical-Sized Instances of Multivariate PKCs: Rainbow, TTS, and lIC-Derivatives
PQCrypto '08 Proceedings of the 2nd International Workshop on Post-Quantum Cryptography
Secure PRNGs from Specialized Polynomial Maps over Any $\mathbb{F}_{q}$
PQCrypto '08 Proceedings of the 2nd International Workshop on Post-Quantum Cryptography
Cryptanalysis of a Multivariate Public Key Encryption Scheme with Internal Perturbation Structure
Information Security Applications
Constructing linear transformations of MPKC by generalized central symmetric matrices
ASID'09 Proceedings of the 3rd international conference on Anti-Counterfeiting, security, and identification in communication
l-invertible cycles for multivariate quadratic (MQ) public key cryptography
PKC'07 Proceedings of the 10th international conference on Practice and theory in public-key cryptography
New differential-algebraic attacks and reparametrization of rainbow
ACNS'08 Proceedings of the 6th international conference on Applied cryptography and network security
On extended algebraic immunity
Designs, Codes and Cryptography
Cryptanalysis of variants of UOV
ISC'06 Proceedings of the 9th international conference on Information Security
Algebraic attacks on clock-controlled cascade ciphers
INDOCRYPT'06 Proceedings of the 7th international conference on Cryptology in India
An analysis of the XSL algorithm
ASIACRYPT'05 Proceedings of the 11th international conference on Theory and Application of Cryptology and Information Security
Building secure tame-like multivariate public-key cryptosystems: the new TTS
ACISP'05 Proceedings of the 10th Australasian conference on Information Security and Privacy
A "medium-field" multivariate public-key encryption scheme
CT-RSA'06 Proceedings of the 2006 The Cryptographers' Track at the RSA conference on Topics in Cryptology
Tractable rational map signature
PKC'05 Proceedings of the 8th international conference on Theory and Practice in Public Key Cryptography
Growth of the ideal generated by a quadratic boolean function
PQCrypto'10 Proceedings of the Third international conference on Post-Quantum Cryptography
Selecting parameters for the rainbow signature scheme
PQCrypto'10 Proceedings of the Third international conference on Post-Quantum Cryptography
Reducing the key size of rainbow using non-commutative rings
CT-RSA'12 Proceedings of the 12th conference on Topics in Cryptology
Public-Key cryptography from new multivariate quadratic assumptions
PKC'12 Proceedings of the 15th international conference on Practice and Theory in Public Key Cryptography
FSE'07 Proceedings of the 14th international conference on Fast Software Encryption
Solving quadratic equations with XL on parallel architectures
CHES'12 Proceedings of the 14th international conference on Cryptographic Hardware and Embedded Systems
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The XL (eXtended Linearization) equation-solving algorithm belongs to the same extended family as the advanced Gröbner Bases methods F4/F5. XL and its relatives may be used as direct attacks against multivariate Public-Key Cryptosystems and as final stages for many “algebraic cryptanalysis” used today. We analyze the applicability and performance of XL and its relatives, particularly for generic systems of equations over medium-sized finite fields. In examining the extended family of Gröbner Bases and XL from theoretical, empirical and practical viewpoints, we add to the general understanding of equation-solving. Moreover, we give rigorous conditions for the successful termination of XL, Gröbner Bases methods and relatives. Thus we have a better grasp of how such algebraic attacks should be applied. We also compute revised security estimates for multivariate cryptosystems. For example, the schemes SFLASHv2 and HFE Challenge 2 are shown to be unbroken by XL variants.