计算多项式逆的算法

Question

我正在寻找一个算法(或代码)来帮助我计算逆多项式,我需要它来实现NTRUEncrypt.一个容易理解的算法是我更喜欢的,有伪代码用于执行此操作,但它们令人困惑且难以实现,而且我无法真正理解伪代码单独的过程.

用于计算截断多项式环的多项式的逆的任何算法？

Answer 1

我为拥有NTRU的安全创新工作,所以我很高兴看到这种兴趣.

IEEE标准1363.1-2008规定了如何使用最新的参数集实现NTRUEncrypt.它给出了以下反转多项式的规范:

师:

输入是a和b,两个多项式,其中b是N-1度,b_N是b的前导系数.输出为q和r,使得a = q*b + r和deg(r)<deg(b).r_d表示度d的r的系数,即r的前导系数.

a)  Set r := a and q := 0
b)  Set u := (b_N)^–1 mod p
c)  While deg r >= N do
  1)    Set d := deg r(X)
  2)    Set v := u × r_d × X^(d–N)
  3)    Set r := r – v × b
  4)    Set q := q + v
d)  Return q, r

这里,r_d是度d的r的系数.

扩展的欧几里德算法:

a)  If b = 0 then return (1, 0, a)
b)  Set u := 1
c)  Set d := a 
d)  Set v1 := 0
e)  Set v3 := b
f)  While v3 ? 0 do
  1)    Use the division algorithm (6.3.3.1) to write d = v3 × q + t3 with deg t3 < deg v3
  2)    Set t1 := u – q × v1
  3)    Set u := v1
  4)    Set d := v3
  5)    Set v1 := t1
  6)    Set v3 := t3
g)  Set v := (d – a × u)/b  [This division is exact, i.e., the remainder is 0]
h)  Return (u, v, d)

反向Z_p,pa prime:

a)  Run the Extended Euclidean Algorithm with input a and (X^N – 1).  Let (u, v, d) be the output, such that a × u + (X^N – 1) × v = d = GCD(a, (X^N – 1)).
b)  If deg d = 0, return b = d^–1 (mod p) × u
c)  Else return FALSE

反向Z_p ^ e /(M(X),pa prime,M(X)一个合适的多项式,如X ^ N-1

a)  Use the Inversion Algorithmto compute a polynomial b(X) ? R[X] that gives an inverse of a(X) in (R/pR)[X]/(M(X)). Return FALSE if the inverse does not exist. [The Inversion Algorithm may be applied here because R/pR is a field, and so (R/pR)[X] is a Euclidean ring.]
b)  Set n = p
c)  While n <= e do
  1)    b(X) = p × b(X) – a(X) × b(X)^2   (mod M(X)), with coefficients computed modulo p^n
  2)    Set n = p × n
d)  Return b(X) mod M(X) with coefficients computed modulo p^e.

如果您正在全面实施NTRU,您应该看看是否可以让您的机构购买1363.1,因为原始NTRU加密对于主动攻击者并不安全,而1363.1描述了解决该问题的消息处理技术.

(更新2013-04-18:感谢Sonel Sharam发现以前版本中的一些错误)