I was wondering if anyone could tell me how to implement line 45 of the following pseudo-code.
Require: the polynomial to invert a(x), N, and q.
1: k = 0
2: b = 1
3: c = 0
4: f = a
5: g = 0 {Steps 5-7 set g(x) = x^N - 1.}
6: g[0] = -1
7: g[N] = 1
8: loop
9: while f[0] = 0 do
10: for i = 1 to N do
11: f[i - 1] = f[i] {f(x) = f(x)/x}
12: c[N + 1 - i] = c[N - i] {c(x) = c(x) * x}
13: end for
14: f[N] = 0
15: c[0] = 0
16: k = k + 1
17: end while
18: if deg(f) = 0 then
19: goto Step 32
20: end if
21: if deg(f) < deg(g) then
22: temp = f {Exchange f and g}
23: f = g
24: g = temp
25: temp = b {Exchange b and c}
26: b = c
27: c = temp
28: end if
29: f = f XOR g
30: b = b XOR c
31: end loop
32: j = 0
33: k = k mod N
34: for i = N - 1 downto 0 do
35: j = i - k
36: if j < 0 then
37: j = j + N
38: end if
39: Fq[j] = b[i]
40: end for
41: v = 2
42: while v < q do
43: v = v * 2
44: StarMultiply(a; Fq; temp;N; v)
45: temp = 2 - temp mod v
46: StarMultiply(Fq; temp; Fq;N; v)
47: end while
48: for i = N - 1 downto 0 do
49: if Fq[i] < 0 then
50: Fq[i] = Fq[i] + q
51: end if
52: end for
53: {Inverse Poly Fq returns the inverse polynomial, Fq, through the argument list.}
The function StarMultiply returns a polynomial (array) stored in the variable temp. Basically temp is a polynomial (I'm representing it as an array) and v is an integer (say 4 or 8), so what exactly does temp = 2-temp mod v equate to in normal language? How should i implement that line in my code. Can someone give me an example.
The above algorithm is for computing Inverse polynomials for NTRUEncrypt key generation. The pseudo-code can be found on page 28 of this document. Thanks in advance.
For each entry in temp, temp[i], set temp[i] = 2-temp[i] mod v.
This should correspond to the "Inverse in Z_p^n" section of my response to Algorithm for computing the inverse of a polynomial.
As I look at it now, I think my answer may not do what it says -- it says "Inverse in Z_p^n" but it looks more like an inverse in Z_2^n. So it should work for p=2 but maybe not for anything else.