Write some Software using C++

Cancelled Posted 7 years ago Paid on delivery
Cancelled Paid on delivery

I am using mathematical software coding here.. hopefully it is the same as C++.

My elliptic curve is ellinit([0,0,0,-3267,45630]*(1+O(13^40))) and point P=[-21,324]*(1+O(13^10)).

I need to add 2 points on this elliptic curve and get the p-adic expansion of it.

For example I am using prime p = 13, so it will be 13-adic expansion. 13-adic expansion means you just need to put *(1+O(13^10)) at the back of the calculation. It is important to have *(1+O(13^10)) everywhere in calculation, if not the numbers will get too big and difficult to solve.

2 points that I want to add on curve is Q=12104mP and R=-2P where m=1,..,12103

For m=1,

then Q=12104P = (ellpow(e,P,12104))*(O(13^40)) = [1 + 13 + 13^2 + 5*13^3 + 7*13^4 + 5*13^5 + 2*13^6 + 5*13^7 + 13^8 + 8*13^9 + O(13^10), 7 + 4*13 + 7*13^2 + 11*13^3 + 5*13^4 + 10*13^5 + 3*13^6 + 7*13^7 + 11*13^8 + 10*13^9 + O(13^10)]

and R=-2P =[51,108] = [12 + 3*13 + O(13^10), 4 + 8*13 + O(13^10)]

so now I need to add these 2 points Q + R, in PARI I use elladd(e,Q*(1+O(13^10)),R*(1+O(13^10)))

and I will get

q=elladd(e,(ellpow(e,P,12104))*(1+O(13^10)),[51,108]*(1+O(13^10)))*(1+O(13^10))

= [12 + 8*13 + 5*13^2 + 10*13^3 + 10*13^4 + 7*13^5 + 7*13^6 + 4*13^7 + 12*13^8 + 4*13^9 + O(13^10), 9 + 3*13 + 6*13^2 + 13^3 + 5*13^4 + 2*13^5 + 2*13^6 + 4*13^7 + 10*13^8 + 6*13^9 + O(13^10)]

and now i define

x=q[1] = 12 + 8*13 + 5*13^2 + 10*13^3 + 10*13^4 + 7*13^5 + 7*13^6 + 4*13^7 + 12*13^8 + 4*13^9 + O(13^10)

y=q[2] = 9 + 3*13 + 6*13^2 + 13^3 + 5*13^4 + 2*13^5 + 2*13^6 + 4*13^7 + 10*13^8 + 6*13^9 + O(13^10)

next I substitute x and y series into my R polynomial

R=(81*x^16 + 40662*x^15 + 14353281*x^14 - 460241028*x^13 - 644722959186*x^12 + 39379675354740*x^11 + 5212980804862026*x^10 - 415546630058854656*x^9 - 8202010485984353739*x^8 + 1396767997483732402758*x^7 - 27550698906220673513787*x^6 - 1044392234943529703379852*x^5 + 60770398462922893831446348*x^4 - 1284453663719469166478575296*x^3 + 14183844641879715988450074288*x^2 - 81800517874945025246941522368*x + 196162341839727571433321441856)- (3240*x^14 + 456840*x^13 + 188268624*x^12 - 45834271200*x^11 - 2435651997264*x^10 + 682353767281968*x^9 - 7053953405575680*x^8 - 2553415737499629216*x^7 + 98906717445152189544*x^6 + 1348117411901578667784*x^5 - 162666175355778441465360*x^4 + 4276857451171442758058304*x^3 - 54456600108308451946891776*x^2 + 350065581968511893813480064*x - 918312303919436410092339456)*y

and I will get final answer :

2 + 4*13 + 3*13^2 + 11*13^3 + 11*13^4 + 9*13^5 + 3*13^7 + 9*13^8 + 8*13^9 + O(13^10)

and the first number in series is a0=2

so now I need to find kronecker(2,13)=-1 (kronecker is the legendre symbol, we did kronecker in our last project using legendre symbol)

Since the answer of kronecker is -1, so I will reject m=1.

So now change m=2 because earlier I said m=1,12103, and do same calculation again, if get kronecker(a0,13), reject m=2 and so on.. my aim is to reject all m from 1,...,12103.

After using p=13, we use other prime numbers until all m is rejected. From my calculation, p=13, 103 and 157 will reject almost all values of m. But I need to use more primes p to reject all m.

If using prime p=103, then need to put *(1+O(103^10)) at the end of all calculation.

If using prime p=157, then change to *(1+O(157^10)).

Any question please feel free to ask me.

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