_:Nacd3f51a5cd64d369d789d6c208001d3 "Pascal" .
"problem MQ" .
"NP" .
.
"chapters" .
"algorithm" .
"relinearization" .
_:Ndef13eb460ad4c98971315eb43227e44 "978-3-540-43168-8" .
"multivariate polynomial equations" .
_:N64d6b2802210448a9438d5d54e669916 .
.
"cryptosystem" .
"characteristic 2" .
_:N4176adbb9c6f45f59d61e4a30dd9eab6 "David" .
.
_:N84e46b8a4a694c9388153618abfb9263 _:Nacd3f51a5cd64d369d789d6c208001d3 .
"scheme" .
_:N7bfe0abd5ee9428faffde9df48a5fd56 .
"Data Format" .
.
"UOV scheme" .
"Information and Computing Sciences" .
_:N71831749da474cceab8c7cf6f6e4bfea "dimensions_id" .
"Courtois" .
_:N77b3370c8e214fc286268b32554e4fed .
_:N7df2e9086a1242b192af05d1046062b8 _:N39d6354f5419447da2cc571b7f4c0faa .
.
"Shamir et" .
"multivariate quadratic equations" .
.
_:Ndef13eb460ad4c98971315eb43227e44 "Public Key Cryptography" .
"CP8 Crypto Lab, SchlumbergerSema, 36-38 rue de la Princesse, BP45, F-78430, Louveciennes Cedex, France" .
_:N66f32bd203f644b98f003940dd0c7c50 .
_:N0a5da4f3d53048ecba9e6dfabcfca123 .
"techniques of relinearization" .
.
.
"large systems" .
"Mathematical Sciences" .
.
"method" .
.
"SFLASH" .
"oil" .
_:N7bfe0abd5ee9428faffde9df48a5fd56 "Jean-Daniel" .
"finite field F." .
"FH Aargau, CH-5210, Windisch" .
"choice" .
_:N0a5da4f3d53048ecba9e6dfabcfca123 .
.
_:Nd7576d130aed4951931c67736eff999f .
_:N64d6b2802210448a9438d5d54e669916 _:N7df2e9086a1242b192af05d1046062b8 .
_:Ndef13eb460ad4c98971315eb43227e44 .
_:N39d6354f5419447da2cc571b7f4c0faa _:N27345feb762a4f21b563bee1bd6a9508 .
"different methods" .
"Nessie call" .
"practice" .
"such underdefined systems" .
_:N77b3370c8e214fc286268b32554e4fed .
_:N39d6354f5419447da2cc571b7f4c0faa .
_:N7bfe0abd5ee9428faffde9df48a5fd56 "Tacier" .
"Solving Underdefined Systems of Multivariate Quadratic Equations" .
.
"exhaustive search" .
"The security of several recent digital signature schemes is based on the difficulty of solving large systems of quadratic multivariate polynomial equations over a finite field F. This problem, sometimes called MQ, is known to be NP-hard. When the number m of equations is equal to the number n of variables, and if n < 15, Gr\u00F6bner base algorithms have been applied to solve MQ. In the overdefined case n \u300A m, the techniques of relinearization and XL, due to A. Shamir et. al., have shown to be successful for solving MQ. In signature schemes, we usually have n \u300B m. For example signature schemes Flash and Sflash submitted to Nessie call for primitives or the UOV scheme published at Eurocrypt 1999. Little is known about the security of such underdefined systems. In this paper, three new and different methods are presented for solving underdefined multivariate systems of quadratic equations. As already shown at Eurocrypt 1999, the problem MQ becomes polynomial when n \u2265 m(m+1) for fields F of characteristic 2. We show that for any field, for about n \u2265 2m/7(m + 1), exponential but quite small in practice, the problem becomes polynomial in n.When n \u2192 m the complexity of all our 3 algorithms tends to qm. However for practical instances of cryptosystems with n \u2248 O(m), we show how to achieve complexities significantly lower than exhaustive search. For example we are able break Unbalanced Oil and Vinegar signature schemes for some \u201Cbad\u201D choices of the parameters (but not for the parameters proposed in [4])." .
"Vinegar (UOV) signature scheme" .
"ET" .
.
"true"^^ .
"underdefined systems" .
_:N71831749da474cceab8c7cf6f6e4bfea .
"Goubin" .
_:N27345feb762a4f21b563bee1bd6a9508 .
.
.
"Eurocrypt 1999" .
"instances" .
"number" .
"en" .
"Gr\u00F6bner base algorithms" .
"quadratic equation" .
_:N4176adbb9c6f45f59d61e4a30dd9eab6 "Naccache" .
"field F" .
_:N66f32bd203f644b98f003940dd0c7c50 _:N84e46b8a4a694c9388153618abfb9263 .
_:N0a5da4f3d53048ecba9e6dfabcfca123 "doi" .
.
"example" .
"multivariate systems" .
.
"polynomial equation" .
"chapter" .
.
.
_:Ndef13eb460ad4c98971315eb43227e44 .
.
"able break Unbalanced Oil" .
_:N0a5da4f3d53048ecba9e6dfabcfca123 "10.1007/3-540-45664-3_15" .
_:Ndef13eb460ad4c98971315eb43227e44 "978-3-540-45664-3" .
_:N71831749da474cceab8c7cf6f6e4bfea "pub.1007371554" .
"field" .
"schemes Flash" .
"variables" .
"practical instances" .
"field F." .
.
"break Unbalanced Oil" .
_:N27345feb762a4f21b563bee1bd6a9508 _:N7bfe0abd5ee9428faffde9df48a5fd56 .
"primitives" .
.
"2021-12-01T20:02" .
_:N4176adbb9c6f45f59d61e4a30dd9eab6 .
"system" .
_:N7df2e9086a1242b192af05d1046062b8 .
"problem" .
"paper" .
_:Nd7576d130aed4951931c67736eff999f "Springer Nature - SN SciGraph project" .
"2002-02-05" .
_:N77b3370c8e214fc286268b32554e4fed "Springer Nature" .
"2002-02-05" .
.
"Louis" .
"digital signature scheme" .
"number n" .
"recent digital signature schemes" .
"difficulties" .