A New Bound between Higher Order Nonlinearity and Algebraic Immunity

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1 Available olie at wwwsciecedirectcom Procedia Egieerig 9 (01) Iteratioal Workshop o Iformatio ad Electroics Egieerig (IWIEE) A New Boud betwee Higher Order Noliearity ad Algebraic Immuity Xueyig Dua a * Departmet of Electroic Egieerig Hua Uiversity of Sciece ad Egieerig Yogzhou 45100, Chia Abstract It is kow that Boolea fuctios used i stream ciphers should have high oliearity to resist algebraic attacks, ad there are some relatioships betwee the algebraic immuity ad the oliearity of a Boolea fuctio Recetly, some researchers have give bouds betwee higher order oliearities ad algebraic immuity However, those bouds are ot very precise I this paper, we try to improve upo them, ad derive a much more precise result o the higher order oliearities of algebraic immue fuctios 011 Published by Elsevier Ltd Selectio ad/or peer-review uder resposibility of Harbi Uiversity of Sciece ad Techology Ope access uder CC BY-NC-ND licese Keywords: Stream ciphers, algebraic immuity, oliearity 1 Itroductio Recetly, algebraic attacks have received a lot of attetio i the cryptographic literature To measure the resistace to algebraic attacks, algebraic immuity has bee proposed by W Meier, et al [4] Noliearity is also a very importat cryptographic criterio to measure the resistace to algebraic attacks, ad a Boolea fuctio with low oliearity will have low algebraic immuity [7] If the algebraic immuity of a fuctio f is kow, the first lower boud o the first order oliearity of f has bee obtaied i [3] I [6], Lobaov improved upo that lower boud I [], a geeralizatio to * Correspodig author Tel: address: duadua169@163com Published by Elsevier Ltd doi:101016/jproeg Ope access uder CC BY-NC-ND licese

2 Xueyig Dua / Procedia Egieerig 9 (01) the r -th order oliearity of the boud obtaied i [3] has bee give, ad i [1], C Carlet improved i some cases upo the correspodig boud of [] I this paper, we deduce some results o algebraic degree ad algebraic immuity of a Boolea fuctio, ad obtai a ew boud which improves upo the boud of [1] ad [] I fact, our boud is two times of the boud of [], ad is greater tha the boud of [1] for ay order r Prelimiaries Let F be the set of all -tuples of elemets i the fiite field F A Boolea fuctio of variables is a fuctio from F ito F We deote by B the set of all -variable Boolea fuctios Ay f B ca be represeted uiquely as a multivariate polyomial i F[ x1,, x] / ( x1 + x1,, x + x), which is called its algebraic ormal form(anf) The algebraic degree of f, deoted by deg( f ), is the degree of this polyomial A Boolea fuctio is affie if there exists o term of degree strictly greater tha 1 i the ANF ad the set of all affie fuctios is deoted by A Let 1 = { x F f( x) = 1},0 = { x F f( x) = 0} f f The cardiality of 1 f, deoted by wt( f ), is called the Hammig weight of f The Hammig distace betwee two fuctios f ad g, deoted by d( f, g ), is the Hammig weight of f + g Let f B The oliearity of f is its distace from the set of all -variable affie fuctios, ie, l( f ) = mig A ( d( f, g)) The Hammig distace betwee f ad the set of Boolea fuctios of algebraic degrees at most r, deoted by lr ( f ), is called r -th order oliearity For ay f B, a ozero fuctio g B is called a aihilator of f if fg= 0, ad the algebraic immuity of f, deoted by AI( f ), is the miimum value of d such that f or f + 1 admits a aihilatig fuctio of degree d As f or 1+ f must have a aihilator of degree, we always have AI ( f ) [4,5] 3 A boud betwee higher order oliearity ad algebraic immuity Lemma 1 Let f, g B d r 1, ad AI( f ) = d, deg ( g ) = r Let T 1 = { x f ( x) = 1 ad g( x) = 0} The

3 790 Xueyig Dua / Procedia Egieerig 9 (01) Proof If r 1 <, the there exists h B of degree at most d r 1 such that hx ( ) = 0 for ay x T1 Therefore, f( g+ 1) h= 0 ad deg(( g + 1) h) = d 1, which is cotradictory to the fact that AI( f) r 1 = d Hece Let T = { x f ( x) = 1 ad g( x) = 1}, T3 = { x f () x = 0 ad g() x = 0}, ad T 4 = { x f ( x) = 0 ad g( x) = 1} I the same way, we ca deduce that r 1 Tk, for k =,3, 4 Particularly, if r = 1, we have d Tk, for k = 1,,3,4 Corollary 1 Let f B, ad AI( f) = d The d d wt( f ) 0 i 0 i Proof Let r = 1 Clearly, wt( f ) = + T, ad wt( f + 1) = T3 + T4 Sice T k d 0 i, for k = 1,,3,4, ad wt( f ) + wt( f + 1) =, we deduce the result Lemma Let f, g B, AI( f) = d ad deg( g) = r < d The d r 1 wt( fg) wt( f ) Proof Let T = { x f ( x) = 1 ad g( x) = 0} Clearly, wt( fg) = wt( f ) T If r 1 T <, the there exist h B of degree at most d r 1 such that hx ( ) = 0 for ay x T Therefore, f( g+ 1) h = 0 ad deg(( g + 1) h) = d 1, which is cotradictory to the fact that AI( f) = d Hece T r 1 0, ad the i result follows Theorem 3 Let f B ad AI( f) = d The d d 1 l( f ) + 0 i 0 i

4 Xueyig Dua / Procedia Egieerig 9 (01) Proof By Lemma 1 ad Lemma, we have d r 1 d( f, g) = wt( f + g) wt( g) + wt( f ) If g A, the wt( g) 1 = ad r = 1 Hece d 1 wt( f + g) + wt( f ) d By Theorem, we have wt( f ), ad the result follows Theorem 4 Let f B of algebraic immuity d ad r be ay positive iteger strictly less tha d The d r 1 lr ( f ) Proof Let g B of algebraic degree at most r Let T1 = { x f ( x) = 1 ad g( x) = 0} ad T = { x f ( x) = 0 ad g( x) = 1} Clearly, d( f, g) = wt( f + g) = + T By Lemma, we have r 1 Similarly, T = { x f ( x ) + 1 = 1 ad g ( x ) + 1 = 0}, where AI( f + 1) = d ad deg( g + 1) r r 1 Hece we have T too, ad the result follows Remarks: 1 The boud of Theorem 4 also holds for r = 1, ad whe it is optimal to the boud of Theorem 3 The boud of Theorem 4 improves upo the boud l ( f ) r of [] ad AI ( f ) r 1 AI ( f ) r 1 r lr ( f ) 0 i of [1] Table 1 THE LOWER BOUNDS ON lr ( f ) give by theorem 4 ad by [1] ad [], for = 11,13,16,19,, r AND AI( f ) OPTIMUM r The boud of Th 4 [1] []

5 79 Xueyig Dua / Procedia Egieerig 9 (01) Refereces [1] Claude Carlet O the Higher Order Noliearities of Algebraic Immue CRYPTO 006, volume 4117 of Lecture Notes i Computer Sciece, pages , Spriger-Verlag, 006 [] Claude Carlet, Deepak Kumar Dalai, Kisha Chad Gupta, Subhamoy Maitra Algebraic Immuity for Cryptographically Sigificat Boolea Fuctios: Aalysis ad Costructio, IEEE Trasactios o IformatioTheory, 006, 5(7): [3] Claude Carlet, Sihem Mesager Improvig the Upper Bouds o the Coverig Radii of Biary Reed-Muller Codes, IEEE Trasactios o Iformatio Theory, 007, 53(1): [4] Willi Meier, Eic Pasalic, Claude Carlet Algebraic attacks ad decompositio of Boolea fuctios, EUROCRYPT 004, volume 307 of Lecture Notes i Computer Sciece, pages Spriger-Verlag, 004 [5] Nicolas Courtois, Willi Meier Algebraic attacks o stream ciphers with liear feedback, EUROCRYPT 003, volume 656 of Lecture Notes i Computer Sciece, pages Spriger-Verlag, 003 [6] Mikhail Lobaov, Tight boud Betwee Noliearity ad algebraic immuity, Paper 005/441 i [7] Qichu Wag, Jie Peg, Haibi Ka, Xiagyag Xue Costructios of Cryptographically Sigificat Boolea Fuctios Usig Primitive Polyomials, IEEE Trasactios o IformatioTheory, 010, 56(6):