Open in CoCalc
R.<u>=QQ['u'] p=(u-1)^2*(u^4-u^2+1)/3+u r=u^4-u^2+1 d=(p^4-p^2+1)/r C.<c>=ZZ['c']
l0=u^5-2*u^4+2*u^2-u+3 l1=u^4-2*u^3+2*u-1 l2=u^3-2*u^2+u l3=u^2-2*u+1
#Our new development of the vectors li (0<=i<=3)---------------- s3=u^2-(u-1)-u s0=l1*u+3 s1=l2*u-l3 s2=s3*u
l0==s0; l1==s1; l2==s2; l3==s3
True True True True
l0^2
u^10 - 4*u^9 + 4*u^8 + 4*u^7 - 10*u^6 + 10*u^5 - 8*u^4 - 4*u^3 + 13*u^2 - 6*u + 9
u=(-2^107+2^105+2^93+2^5) p=(u-1)^2*(u^4-u^2+1)/3+u r=u^4-u^2+1 t=u+1 #-------------- Fp-------------- Fp=GF(p) #------------------------------- b=2 #--------------Fp^2-------------- K2.<i>=PolynomialRing(Fp) Fp2.<i>=FiniteField(p^2,modulus=i^2+1) #------------------------------------ #-------------Fp^6--------------------- K6.<v>=PolynomialRing(Fp2) Fp6.<v>=Fp2.extension(v^3-i-1) #------------------------------------- #---------------Fp^12------------------- K12.<z>=PolynomialRing(Fp6) Fp12.<z>=Fp6.extension(z^2-v) Fp12.is_field=lambda:True #---------------------------------------
147bb7b-824f-4aba-b95b-10ab277f807es Fp12.is_field(); Fp6.is_field(); Fp2.is_field();
f1=Fp12.random_element()
Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'Fp12' is not defined
f=(f1^(p^6-1))^(p^2+1) #algorithm 1: New variant of Aranha et al. for computing the hard part of the final exponentiation of optimal Ate pairing on BLS12 curves def algorithm_1 (f,u): t0=f^2 t1=t0^(u) t2=t1^(u/2) t3=f^(-1) t1=t3*t1 t1=t1^(-1) t1=t1*t2 t2=t1^(u) t3=t2^(u) t1=t1^(-1) t3=t1*t3 t1=t1^(-1) t1=t1^(p^3) t2=t2^(p^2) t1=t1*t2 t2=t3^(u) t2=t2*t0 t2=t2*f t1=t1*t2 t2=t3^p t1=t1*t2 return t1
Error in lines 1-2 Traceback (most recent call last): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'f1' is not defined
algorithm_1 (f,u)==f^(3*(p^4-p^2+1)/r)
Error in lines 1-1 Traceback (most recent call last): File "/projects/sage/sage-6.10/local/lib/python2.7/site-packages/smc_sagews/sage_server.py", line 905, in execute exec compile(block+'\n', '', 'single') in namespace, locals File "", line 1, in <module> NameError: name 'algorithm_1' is not defined