{"id":367,"date":"2018-04-23T18:47:23","date_gmt":"2018-04-23T09:47:23","guid":{"rendered":"http:\/\/tamatoyaku.com\/b\/?p=367"},"modified":"2018-04-23T18:47:23","modified_gmt":"2018-04-23T09:47:23","slug":"367","status":"publish","type":"post","link":"https:\/\/p-0.me\/b\/p\/367\/","title":{"rendered":"PARI GP\u3067Montgomery curves\u3092\u7528\u3044\u305fECDSA\u306e\u5b9f\u88c5(2)"},"content":{"rendered":"<p><a href=\"https:\/\/tamatoyaku.com\/b\/p\/365\">\u524d\u56de<\/a>\u4f5c\u6210\u3057\u305f\u30b3\u30fc\u30c9\u3067\u306fWeierstrass\u6a19\u6e96\u5f62\u3068Montgomery curves\u3092\u540c\u6642\u306b\u5b9f\u884c\u3057\u305f\uff0e\u4eca\u56de\u306fMontgomery curves\u306e\u307f\u3067\u5b9f\u884c\u3057\u3066\u307f\u308b\uff0e<br \/>\n<!--more--><\/p>\n<pre class=\"lang:default decode:true \">\\\\m-n==1\nadd(P1,Pm,Pn,p)={\n\tlocal(a,b,c,d,Xmn,Zmn);\n\tif(Pm==0||Pm[2]==0,return(Pn));\n\tif(Pn==0||Pn[2]==0,return(Pm));\n\ta=Pm[1]+Pm[2];\n\tb=Pm[1]-Pm[2];\n\tc=Pn[1]+Pn[2];\n\td=Pn[1]-Pn[2];\n\tXmn=Mod(P1[2]*((b*c+a*d)^2),p);\n\tZmn=Mod(P1[1]*((b*c-a*d)^2),p);\n\treturn([Xmn,Zmn]);\n}\ndbl(A,P,p)={\n\tlocal(tmp1,tmp2,tmp3,X2,Z2);\n\tif(P==0||P[2]==0,return([0]));\n\ttmp1=(P[1]+P[2])^2;\n\ttmp2=(P[1]-P[2])^2;\n\ttmp3=tmp1-tmp2;\n\tX2=Mod(tmp1*tmp2,p);\n\tZ2=Mod(tmp3*(tmp2+((tmp3*(A+2))\/4)),p);\n\treturn([X2,Z2]);\n}\nrecover(A,B,x1,y1,xn,xn1,p)={\n\tlocal(up,down,res);\n\tif(xn1==-1,\n\t\treturn(-y1);\n\t);\n\tup=(x1*xn+1)*(x1+xn+2*A)-2*A-((x1-xn)^2)*xn1;\n\tdown=2*B*y1;\n\tif(down==0,\n\t\tres=0;,\n\t\tres=Mod(up\/down,p);\n\t);\n\treturn(res);\n}\nladder(A,P,k,p)={\n\tlocal(R0,R1,i,bin);\n\tbin=binary(k);\n\tR0=[0];\n\tR1=P;\n\tfor(i=1,length(bin),\n\t\tif(bin[i]==0,\n\t\t\tR1=add(P,R1,R0,p);\n\t\t\tR0=dbl(A,R0,p);,\n\t\t\tR0=add(P,R1,R0,p);\n\t\t\tR1=dbl(A,R1,p);\n\t\t);\n\t);\n\treturn([R1,R0]);\n}\n\\\\input and output is affine point P=(x,y)\nscalar(A,B,P,k,p)={\n\tlocal(P_M,pair,kP_M,k1P_M,x_k,x_k1,y_k);\n\tP_M=[P[1],1];\n\tpair=ladder(A,P_M,k,p);\n\tkP_M=pair[2];k1P_M=pair[1];\n\tif(kP_M!=[0]&amp;&amp;kP_M[2]==0,kP_M=[0]);\n\tif(k1P_M!=[0]&amp;&amp;k1P_M[2]==0,k1P_M=[0]);\n\tif(kP_M==[0],\n\t\treturn([0]);\n\t);\n\tif(kP_M!=[0],\n\t\tx_k=Mod(kP_M[1]\/kP_M[2],p);\n\t\tif(k1P_M==[0]||k1P_M[2]==0,\n\t\t\tx_k1=-1;,\n\t\t\tx_k1=Mod(k1P_M[1]\/k1P_M[2],p);\n\t\t);\n\t\ty_k=recover(A,B,P_M[1],P[2],x_k,x_k1,p);\n\t);\n\treturn([x_k,y_k]);\n}\nmdbl(A,B,P,p)={\n\tlocal(x,y);\n\tx=Mod(((P[1]^2-1)^2)\/(4*B*(P[2]^2)),p);\n\ty=Mod(((2*P[1]+P[1]+A)*(3*(P[1]^2)+2*A*P[1]+1))\/(2*B*P[2])-(B*(3*(((P[1]^2)+2*A*P[1]+1)^3)))\/((2*B*P[2])^3)-P[2],p);\n\treturn([x,y]);\n}\nmadd(A,B,P1,P2,p)={\n\tlocal(x,y);\n\tif(P1==[0],return(P2));\n\tif(P2==[0],return(P1));\n\tif(P1[1]==P2[1],\n\t\tif(P1[2]==-P2[2],\n\t\t\treturn([0]);,\n\t\t\treturn(mdbl(A,B,P1,p));\n\t\t);\n\t);\n\tx=Mod((B*((P2[2]-P1[2])^2))\/((P2[1]-P1[1])^2)-A-P1[1]-P2[1],p);\n\ty=Mod(((2*P1[1]+P2[1]+A)*(P2[2]-P1[2]))\/(P2[1]-P1[1])-(B*((P2[2]-P1[2])^3))\/((P2[1]-P1[1])^3)-P1[2],p);\n\treturn([x,y]);\n}\n{\n\t\\\\\u6e96\u5099\n\tA=84;B=1;p=30011;\n\tP=[Mod(7768,p),Mod(9454,p)];\n\tl=1249;\n\tprint(\"pripare\");\n\tprint(\"\\tbase point P=\",P);\n\tprint(\"\\torder l=\",l,\"\\n\");\n\t\\\\\u9375\u751f\u6210\n\td_B=random(l-2)+2;\n\tP_B=scalar(A,B,P,d_B,p);\n\tprint(\"key generation\");\n\tprint(\"\\tsecret key d_B =\",d_B);\n\tprint(\"\\tpublic key P_B =\",P_B,\"\\n\");\n\t\\\\\u7f72\u540d\u751f\u6210\n\tuntil(u!=0&amp;&amp;v!=0,\n\t\tr=random(l-1)+1;\n\t\tU=scalar(A,B,P,r,p);\n\t\tm=random(p);\n\t\tu=lift(lift(U[1])%l);\n\t\tv=((m+u*d_B)\/r)%l;\n\t);\n\tprint(\"signature generation\");\n\tprint(\"\\trandom value r=\",r);\n\tprint(\"\\tU=\",U);\n\tprint(\"\\tm=\",m);\n\tprint(\"\\tsignature (u,v)=(\",u,\",\",v,\")\",\"\\n\");\n\t\\\\\u7f72\u540d\u691c\u8a3c\n\td=(1\/v)%l;\n\tV=madd(A,B,scalar(A,B,P,d*m,p),scalar(A,B,P_B,d*u,p),p);\n\tprint(\"signature verification\");\n\tprint(\"\\td=\",d);\n\tprint(\"\\tV=\",V,\"\\n\");\n\t\\\\\u7d50\u679c\u8868\u793a\n\tprint(\"result\");\n\tprint(\"\\t(u,Vx)=(\",u,\",\",lift(V[1])%l,\")\");\n\tif(u==(lift(V[1])%l),\n\t\tprint(\"\\tOK\");,\n\t\tprint(\"\\tNG\");\n\t);\n}\n<\/pre>\n<p>&nbsp;<\/p>\n<pre class=\"lang:default decode:true \">(18:47) gp &gt; \\r m_ecdsa.gp\npripare\n        base point P=[Mod(7768, 30011), Mod(9454, 30011)]\n        order l=1249\nkey generation\n        secret key d_B =1047\n        public key P_B =[Mod(24356, 30011), Mod(3467, 30011)]\nsignature generation\n        random value r=40\n        U=[Mod(28450, 30011), Mod(25680, 30011)]\n        m=28429\n        signature (u,v)=(972,642)\nsignature verification\n        d=1142\n        V=[Mod(28450, 30011), Mod(25680, 30011)]\nresult\n        (u,Vx)=(972,972)\n        OK<\/pre>\n<p>&nbsp;<br \/>\n&nbsp;<\/p>\n","protected":false},"excerpt":{"rendered":"<p>\u524d\u56de\u4f5c\u6210\u3057\u305f\u30b3\u30fc\u30c9\u3067\u306fWeierstrass\u6a19\u6e96\u5f62\u3068Montgomery curves\u3092\u540c\u6642\u306b\u5b9f\u884c\u3057\u305f\uff0e\u4eca\u56de\u306fMontgomery curves\u306e\u307f\u3067\u5b9f\u884c\u3057\u3066\u307f\u308b\uff0e<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[4],"tags":[],"class_list":["post-367","post","type-post","status-publish","format-standard","hentry","category-4"],"_links":{"self":[{"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/posts\/367","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/comments?post=367"}],"version-history":[{"count":0,"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/posts\/367\/revisions"}],"wp:attachment":[{"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/media?parent=367"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/categories?post=367"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/p-0.me\/b\/wp-json\/wp\/v2\/tags?post=367"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}