Idempotent and p-potent quadratic functions: distribution of nonlinearity and co-dimension

Abstract

The Walsh transform $\widehat{Q}$ of a quadratic function $Q:{\mathbb F}_{p^n}\rightarrow {\mathbb F}_p$ satisfies $|\widehat{Q}(b)| \in \{0,p^{\frac{n+s}{2}}\}$ for all $b\in {\mathbb F}_{p^n}$, where $0\le s\le n-1$ is an integer depending on Q. In this article, we study the following three classes of quadratic functions of wide interest. The class $\mathcal {C}_1$ is defined for arbitrary n as $\mathcal {C}_1 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{\lfloor (n-1)/2\rfloor }a_ix^{2^i+1})\;:\; a_i \in {\mathbb F}_2\}$, and the larger class $\mathcal {C}_2$ is defined for even n as $\mathcal {C}_2 = \{Q(x) = \mathrm{Tr_n}(\sum _{i=1}^{(n/2)-1}a_ix^{2^i+1}) + \mathrm{Tr_{n/2}}(a_{n/2}x^{2^{n/2}+1}) \;:\; a_i \in {\mathbb F}_2\}$. For an odd prime p, the subclass $\mathcal {D}$ of all p-ary quadratic functions is defined as $\mathcal {D} = \{Q(x) = \mathrm{Tr_n}(\sum _{i=0}^{\lfloor n/2\rfloor }a_ix^{p^i+1})\;:\; a_i \in {\mathbb F}_p\}$. We determine the generating function for the distribution of the parameter s for $\mathcal {C}_1, \mathcal {C}_2$ and $\mathcal {D}$. As a consequence we completely describe the distribution of the nonlinearity for the rotation symmetric quadratic Boolean functions, and in the case $p > 2$, the distribution of the co-dimension for the rotation symmetric quadratic p-ary functions, which have been attracting considerable attention recently. Our results also facilitate obtaining closed formulas for the number of such quadratic functions with prescribed s for small values of s, and hence extend earlier results on this topic. We also present the complete weight distribution of the subcodes of the second order Reed–Muller codes corresponding to $\mathcal {C}_1$ and $\mathcal {C}_2$ in terms of a generating function.

Appendix

We give some examples of generating functions for $p=3$.

Example

$n=9\cdot 13$: In this case, $x^{9\cdot 13}-1 = (x-1)^9r_1^9r_2^9$ for prime self-reciprocal polynomials $r_1,r_2$ both of degree 6. By Theorem 3,

$$\begin{aligned} \mathcal {G}_{9\cdot 13}^{(3)}(z) = \left( 1+2\sum _{j=1}^53^{j-1}z^{2j-1}\right) \left( 1+26\sum _{j=1}^93^{3(j-1)}z^{6j}\right) ^2. \end{aligned}$$

Expanding this polynomial we obtain

$$\begin{aligned}&\mathcal {G}_{9\cdot 13}^{(3)}(z)\\&\quad = 1+2\,z+6\,{z}^{3}+18\,{z}^{5}+52\,{z}^{6}+158\,{z}^{7}+474\,{z}^{9}+936\,{z}^{11}+2080\,{z}^{12}+6968\,{z}^{13}\\&\qquad +20904\,{z}^{15}+37440\,{z}^{17} +74412\,{z}^{18}+261144\,{z}^{19}+783432\,{z}^{21}+1339416\,{z}^{23}\\&\qquad +2501928\,{z}^{24}+9022104\,{z}^{25}+27066312\,{z}^{27}+45034704\,{z}^{29}+80857764\,{z}^{30}\\&\qquad +296819640\,{z}^{31}+890458920\,{z}^{33}+1455439752\,{z}^{35}+2542413744\,{z}^{36}\\&\qquad +9451146744\,{z}^{37}+28353440232\,{z}^{39}+45763447392\,{z}^{41}+78345032220\,{z}^{42}\\&\qquad +293980406616\,{z}^{43}+881941219848\,{z}^{45} +1410210579960\,{z}^{47}\\&\qquad +2377212120504\,{z}^{48}+8985055980888\,{z}^{49}+26955167942664\,{z}^{51}\\&\qquad +42789818169072\,{z}^{53}+71255926018836\,{z}^{54}+270881306544888\,{z}^{55}\\&\qquad +812643919634664\,{z}^{57}+1282606668339048\,{z}^{59}+1718301299950404\,{z}^{60}\\&\qquad + 7284422604917952\,{z}^{61}+21853267814753856\,{z}^{63}+30929423399107272\,{z}^{65}\\&\qquad +41239231198809696\,{z}^{66}+175266732594941208\,{z}^{67}+525800197784823624\,{z}^{69}\\ \end{aligned}$$

$$\begin{aligned}&\qquad +742306161578574528\,{z}^{71} +974276837071879068\,{z}^{72}+4175472158879481720\,{z}^{73}\\&\qquad +12526416476638445160\,{z}^{75}+17536983067293823224\,{z}^{77}\\&\qquad +22547549657949201288\,{z}^{78}+97706048517779872248\,{z}^{79}\\&\qquad +293118145553339616744\,{z}^{81}+405855893843085623184\,{z}^{83}\\&\qquad +507319867303857028980\,{z}^{84}+2232207416136970927512\,{z}^{85}\\&\qquad +6696622248410912782536\,{z}^{87}+9131757611469426521640\,{z}^{89}\\&\qquad +10958109133763311825968\,{z}^{90}+49311491101934903216856\,{z}^{91}\\&\qquad +147934473305804709650568\,{z}^{93} +197245964407739612867424\,{z}^{95}\\&\qquad +221901709958707064475852\,{z}^{96}+1035541313140632967553976\,{z}^{97}\\&\qquad +3106623939421898902661928\,{z}^{99} +3994230779256727160565336\,{z}^{101}\\&\qquad +3994230779256727160565336\,{z}^{102}+19971153896283635802826680\,{z}^{103}\\&\qquad +59913461688850907408480040\,{z}^{105}+71896154026621088890176048\,{z}^{107}\\&\qquad +53922115519965816667632036\,{z}^{108} +323532693119794900005792216\,{z}^{109}\\&\qquad +970598079359384700017376648\,{z}^{111}+970598079359384700017376648\,{z}^{113}\\&\qquad +2911794238078154100052129944\,{z}^{115} +8735382714234462300156389832\,{z}^{117} \end{aligned}$$

Example

$n=9\cdot 14$: In this case, $x^{9\cdot 14}-1 = (x-1)^9(x+1)^9r_1^9r_2^9$ for prime self-reciprocal polynomials $r_1,r_2$ both of degree 6. By Theorem 4,

$$\begin{aligned}&\mathcal {G}_{9\cdot 14}^{(3)}(z) \\&\quad = \left( 1+2\sum _{j=1}^53^{j-1}z^{2j-1}\right) ^2\left( 1+26\sum _{j=1}^93^{3(j-1)}z^{6j}\right) ^2 \\&\quad = 1+4\,z+4\,{z}^{2}+12\,{z}^{3}+24\,{z}^{4}+36\,{z}^{5}+160\,{z}^{6}+316\,{z}^{7}+640\,{z}^{8}+948\,{z}^{9}+2868\,{z}^{10}\\&\qquad +1872\,{z}^{11}+11584\,{z}^{12}+13936\,{z}^{13}+39532\,{z}^{14}+41808\,{z}^{15}+151656\,{z}^{16}+74880\,{z}^{17}\\&\qquad +527472\,{z}^{18}+522288\,{z}^{19}+1651104\,{z}^{20}+1566864\,{z}^{21}+6065280\,{z}^{22}+2678832\,{z}^{23}\\&\qquad +19990152\,{z}^{24}+18044208\,{z}^{25}+60349536\,{z}^{26}+54132624\,{z}^{27}+216985392\,{z}^{28}\\&\qquad +90069408\,{z}^{29}+694967364\,{z}^{30}+593639280\,{z}^{31}+2055220128\,{z}^{32}\\&\qquad +1780917840\,{z}^{33}+7295622048\,{z}^{34}+2910879504\,{z}^{35}+22955416848\,{z}^{36}\nonumber \\&\qquad +18902293488\,{z}^{37}+66987075168\,{z}^{38}+56706880464\,{z}^{39}+235781239824\,{z}^{40}\\&\qquad +91526894784\,{z}^{41}+732961301436\,{z}^{42}+587960813232\,{z}^{43}+2119046585760\,{z}^{44}\\&\qquad +1763882439696\,{z}^{45}+7413678477504\,{z}^{46}+2820421159920\,{z}^{47}\\&\qquad +22845411395352\,{z}^{48}+17970111961776\,{z}^{49}+65594937833568\,{z}^{50}\\&\qquad +53910335885328\,{z}^{51}+228454113953520\,{z}^{52}+85579636338144\,{z}^{53}\\&\qquad +699323426602164\,{z}^{54}+541762613089776\,{z}^{55}+1997341681993632\,{z}^{56}\\&\qquad +1625287839269328\,{z}^{57}+6931950543389664\,{z}^{58}+2565213336678096\,{z}^{59}\\&\qquad +20712629060085924\,{z}^{60}+14568845209835904\,{z}^{61}+58451616870107760\,{z}^{62}\\ \end{aligned}$$

$$\begin{aligned}&\qquad +43706535629507712\,{z}^{63}+198265534609662000\,{z}^{64}+61858846798214544\,{z}^{65}\\&\qquad +566246366845194672\,{z}^{66}+350533465189882416\,{z}^{67}\\&\qquad +1530609927186590640\,{z}^{68}+1051600395569647248\,{z}^{69}\\&\qquad +5020083336316641840\,{z}^{70}+1484612323157149056\,{z}^{71}\\&\qquad +13978909783188828972\,{z}^{72}+8350944317758963440\,{z}^{73}\\&\qquad +36744154998139439136\,{z}^{74}+25052832953276890320\,{z}^{75}\\&\qquad +120253598175729073536\,{z}^{76}+35073966134587646448\,{z}^{77}\\&\qquad +333202678278582641256\,{z}^{78}+195412097035559744496\,{z}^{79}\\&\qquad +871838586774035783136\,{z}^{80}+586236291106679233488\,{z}^{81}\\&\qquad +2840991256901599362288\,{z}^{82}+811711787686171246368\,{z}^{83}\\&\qquad +7812725956479398246292\,{z}^{84}+4464414832273941855024\,{z}^{85}\\&\qquad +20292794692154281159200\,{z}^{86}+13393244496821825565072\,{z}^{87}\\&\qquad +65748654802579870955808\,{z}^{88}+18263515222938853043280\,{z}^{89}\\&\qquad +178982449184800759824144\,{z}^{90}+98622982203869806433712\,{z}^{91}\\&\qquad +460240583618059096690656\,{z}^{92}+295868946611609419301136\,{z}^{93}\\&\qquad +1479344733058047096505680\,{z}^{94}+394491928815479225734848\,{z}^{95}\\ \end{aligned}$$

$$\begin{aligned}&\qquad +3969575033705759708956908\,{z}^{96}+2071082626281265935107952\,{z}^{97}\\&\qquad +10059544184794720256238624\,{z}^{98}+6213247878843797805323856\,{z}^{99}\\&\qquad +31953846234053817284522688\,{z}^{100}+7988461558513454321130672\,{z}^{101}\\&\qquad +83878846364391270371872056\,{z}^{102}+39942307792567271605653360\,{z}^{103}\\&\qquad +207700000521349812349397472\,{z}^{104}+119826923377701814816960080\,{z}^{105}\\&\qquad +647065386239589800011584432\,{z}^{106}+143792308053242177780352096\,{z}^{107}\\&\qquad +1635637504105629772251505092\,{z}^{108}+647065386239589800011584432\,{z}^{109}\\&\qquad +3882392317437538800069506592\,{z}^{110}+1941196158718769400034753296\,{z}^{111}\\&\qquad +11647176952312616400208519776\,{z}^{112}+1941196158718769400034753296\,{z}^{113}\\&\qquad +27176746222062771600486546144\,{z}^{114}+5823588476156308200104259888\,{z}^{115}\\&\qquad +58235884761563082001042598880\,{z}^{116}+17470765428468924600312779664\,{z}^{117}\\&\qquad +157236888856220321402815016976\,{z}^{118}\\&\qquad +314473777712440642805630033952\,{z}^{120}\\&\qquad +471710666568660964208445050928\,{z}^{122}\\&\qquad +943421333137321928416890101856\,{z}^{124}\\&\qquad +1415131999705982892625335152784\,{z}^{126} \end{aligned}$$

Example

$n=9\cdot 20$: With $x^{9\cdot 14}-1 = (x-1)^9(x+1)^9r_1^9r_2^9r_3^9,r_4^9$, where $r_1,r_2,r_3,r_4$ are prime self-reciprocal polynomials of degrees 2, 4, 4 and 8, from Theorem 4 we obtain

$$\begin{aligned} \mathcal {G}_{9\cdot 20}^{(3)}(z)&= \left( 1+2\sum _{j=1}^53^{j-1}z^{2j-1}\right) ^2\left( 1+2\sum _{j=1}^93^{j-1}z^{2j}\right) \\&\times \left( 1+8\sum _{j=1}^93^{2(j-1)}z^{4j}\right) ^2\left( 1+80\sum _{j=1}^93^{4(j-1)}z^{8j}\right) \end{aligned}$$

Expanding, for instance from the coefficient of $z^{99}$ we see that the number of 81-plateaued functions in $\mathcal {D}$ for $p=3$ and $n=9\cdot 20$ is 616946472137940526877139072. Furthermore we see that the number of bent functions in the set $\mathcal {D}$ is $\mathcal {N}_{9\cdot 20}^{(3)} = 6054249652811609019026768290053459869736960$. Here we omit writing down the whole expanded version of the polynomial $\mathcal {G}_{9\cdot 20}^{(3)}$.