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Multivariate Signature Scheme using Quadratic Forms Takanori Yasuda (ISIT) Joint work with Tsuyoshi Takagi (Kyushu Univ.), Kouichi Sakurai (Kyushu Univ.) This work was partially supported by the Japan Science and Technology Agency (JST) Strategic Japanese-Indian Cooperative Programme for Multidisciplinary Research Fields, which aims to combine Information and Communications Technology with Other Fields. The first author is supported by Grant-in-Aid for Young Scientists (B), Grant number 24740078. 1 Contents 1. 2. 3. 4. 5. Multivariate Signature Schemes Quadratic Forms Multivariate System defined by Quadratic Forms Application to Signature Scheme Comparison with Rainbow 1. 2. 3. Efficiency of Signature Generation Key Sizes Security 6. Conclusion 2 MPKC Signature : → : multivariate polynomial map Vector space Vector space = − () Signature Inverse function −1 Message For any message M, there must exist the corresponding signature. F is surjective. 3 New Multivariate Polynomial Map • We introduce a multivariate polynomial map not surjective, and apply it to signature scheme. Multivariate polynomial map For a symmetric matrix A, () = . . where = is a matrix of variables of size × . is a map which assigns a matrix to a matrix. G can be regarded as 2 2 a multivariate polynomial map → . (+1)/2 4 Questions Is G applicable to signature scheme or not? Questions 1. Can its inverse map be computed efficiently? Necessary to compute −1 M for a message M in order to generate a signature. 2. Is it surjective or not? For any message M, necessary to generate its signature. 5 Quadratic Forms • Definition 1 : Field with odd characteristic (or 0) : Natural number : → is a quadratic form = . . for some symmetric matrix • Definition 2 , : quadratic forms associated to , and are isometric . . = for some ∈ (, ) 6 Translation of questions of in terms of quadratic form • Equation (, : symmetric matrices) () = . . = • Restrict solution ∈ (, ) o Problem 1 For , , isometric each other, find a translation matrix efficiently. o Problem 2 For any , , determine whether and isometric or not? 7 How to compute the inverse map Simple case 1 = = 0 ⋱ 0 1 Problem 1 is equivalent to Problem 1’: Find an orthonormal basis of with respect to . Orthonormal basis: 1 , … in = 1 for = 1, … , , , ≔ . . = 0 for ≠ 8 Real field Case • = : real field Gram-Schmidt orthonormalization provides an efficient algorithm to solve Problem 1’. It uses special property of = . Fact: = is anisotropic. Definition: A quadratic form is anisotropic for any (≠ 0) , () ≠ 0 We want to apply Gram-Schmidt orthonormalization technique to the case of finite fields. 9 Finite Field Case Fact Let be a finite field. Any quadratic form on ( ≥ 3) is not anisotropic. We cannot apply Gram-Schmidt orthonormalization directly. • However, we can extend Gram-Schmidt orthonormalization by inserting a step: If = 0, then find another element ′ such that ′ ≠ 0. Solve Problem 1 10 2-dimensional case (1) Operation for Matrices of 2×2 is fundamental. (1) = ( ≠ 0) In this case, apply the usual GS orthonormalization. 11 2-dimensional case (2) (2) = ( = 0) • There are two cases: = 0 or ≠ 0. ⇒ apply the usual GS-normalization. 12 2-dimensional case (3) • We obtained • 0 ′ = (, ≠ 0) 0 There is a matrix such that 1 0 . ’. = . 0 ′ This completes the Extended GS-normalization. 13 Problem 2 • Definition : quadratic form associated to . is nondegenerate det() ≠ 0 Classification theorem Any nondegenerate quadratic form is isometric to either 1 or . 14 Classification Theorem • For any (nondegenerate) message , either • • • • ∙ 1 ∙ = or ∙ ∙ = has a solution. 1 or is determined by det(). In the degenerate case, both equations have solutions. = ∙ 1 ∙ or = ∙ ∙ is not surjective. However, we can apply these maps to MPKC signature scheme. 15 Application to MPKC Signature Scheme • Secret Key 1 , (, ) 1 ≔ 1 . 1 . 1 , ≔ . . , 1 = . 1 . , : → , • Public Key = . . : → , affine transformations = +1 2 , = 2 1 : → defined by 1 = °1 °, : → defined by = ° °, 16 Signature Generation For a symmetric matrix , • Step 1 Compute ’ = −1 () . • Step 2 Apply the extended Gram-Schmidt orthornormalization to ′. o Find a solution = of either ∙ 1 ∙ = ′ or ∙ ∙ = ′ • Step 3 Compute = 1 −1 . or = −1 . . = is a solution of 1 = or = . • Step 4 Compute = −1 (). 17 Property of Our Scheme • Respective map 1 or is not surjective. • However, the union of images of these maps covers the whole space. For any M, there exists the corresponding signature. M 18 Other Signature Schemes Multivariate Polynomial Maps Rainbow Surjective HFE UOV MI Not Surjective Proposal 19 Security of Our Scheme • There are several attacks of MPKC signature schemes which depend on the structure of central map. • For example, UOV attack is an attack which transforms public key into a form of central map of UOV scheme. o Central maps of UOV are surjective. o The public key of our scheme cannot be transformed into any surjective map. • These attacks is not applicable against our scheme. （Other examples: Rainbow-band-separation attack, UOV-Reconciliation attack） • However, attacks which is independent of scheme, like direct attacks, are applicable to our scheme. 20 Comparison with Rainbow Compared in the case that and are same for public key F : → • Equivalent with respect to cost of verification and public key length. • Cost of signature generation (number of mult.) o Proposal (2 ) o Rainbow (3 ) ⇒ 8 or 9 times more efficient at the level of 88-bit security. • Secret Key Size (number of elements of field) o Proposal o Rainbow 21 Conclusion • We propose a new MPKC signature scheme using quadratic forms. The multivariate polynomial map used in the scheme is not surjective. • Signature generation uses an extended Gram-Schmidt orthonormalization. It is 8 or 9 times more efficient than that of Rainbow at the level of 88-bit security. Future Work • Security analysis • Application to encryption scheme 22