帮我写一段N值较小的用基础函数实现的NTRU数字签名功能的MATLAB代码，并测试代码

好的，我可以为您写一段NTRU数字签名功能的MATLAB代码，不过在此之前我需要了解一些关于NTRU数字签名的知识，以便于更好地进行编码设计和测试。

NTRU数字签名是一种基于格的密码学签名方案，它是通过对一个向量多项式的系数进行转换和控制，来实现数字签名的安全性和效率。该方案使用了R-LWE（Ring-Learning with Errors）问题，这种问题是基于格的加密中的一种难题，由此实现了数字签名的安全性。

针对您的要求，我参考了已有的NTRU数字签名相关代码进行编写，以下是MATLAB代码，包括了密钥生成、签名过程和验证过程：

function [pk, sk] = ntru_keygen(N, p, q)
% N: poly length
% p: plaintext modulus
% q: ciphertext modulus
% holds: holds coefficients of private key polynomial
% f: public key

% generate Gaussian random polynomials f and g
f = generate_poly(N, p);
g = generate_poly(N, p);

% calculate inverse of g
[~,~,inv_g] = poly_gcd_ring(g,p);

% calculate private key
holds = mod(conv(f, inv_g), p);
sk = {holds, g};

% calculate public key
f_q = mod(conv(f,g),q);
pk = {f_q, q};
end

function ret = generate_poly(N, p)
% Generates random polynomial mod p with coefficients in [-p/2, p/2].
ret = randi(p,N,1)-floor(p/2)';  
end

function [gcd, x, y] = poly_gcd_ring(a, p)
% compute gcd of the polynomials a and x^p - x in the ring Zp[x]
b = [1, zeros(1, p-1), -1];
[~, r] = poly_div_ring(a, b, p);
if sum(r(1:p)) ~= 0 || numel(r) ~= p
    error('In poly_gcd_ring: unexpected result.');
end

[p,q,~,gcd] = poly_ext_euclid_ring(a, b, p);

i = find(sign(gcd) < 0, 1);
gcd = mod(gcd, p);
if ~isempty(i) && ~isempty(gcd) && sign(gcd(i)) < 0
    gcd = -gcd;
    q = -q;
end

x = q(:,1);
y = q(:,2);
end

function [q,r] = poly_div_ring(a,b,p)
% polynomial division in the ring Zp[x]
% Returns the quotient q and residual r, such that a = b*q + r]
while numel(a) >= numel(b) && ~all(a==0) %a is longer than b
    shift = numel(a) - numel(b);
    a = mod(a, p);
    leading_coeff = mod(b(1) * modinv(b(1), p), p);
    a_coeff = mod(a(1) * modinv(b(1), p), p);
    quotient_coeff = mod(a_coeff * leading_coeff, p);
    quotient = [zeros(1,shift-1), quotient_coeff];
    b_shifted = [zeros(1, shift), b];
    a = mod(a - mod(conv(b_shifted, quotient), p), p);
end
q = a;
r = mod(a, p);
end

function [p_qinv, qinv] = poly_ext_euclid_ring(a, b, p)
% extended GCD in the polynomial ring Zp[x]
% Given two polynomials a, b and a prime p, returns the Bezout coefficients p and q.

% initialization
p_qinv = {...
    {1, zeros(1, numel(b)-1)};
    {zeros(1, numel(b)-1), 1};
    };
r = {a; b};

while sum(r{end} ~= 0)
    q = poly_div_ring(r{end-1}, r{end}, p);
    r = [r, q];
    p_qinv = [p_qinv, {poly_sub_ring(p_qinv{end-1}, poly_mul_ring(q, p_qinv{end}), p)}];
end

qinv = p_qinv{end-1};
end

function ret = modinv(value, modulus)
% calculate inverse modulo modulus
[~, r, s] = gcd(value, modulus);
if r < 0
    r = r + modulus;
end
ret = mod(s, modulus);

if mod(value*ret,modulus) ~=1
    error('In modinv: computing incorrect result.');
end
end

function ret = poly_sub_ring(a, b, p)
% polynomial subtraction in the ring Zp[x]
% Returns [a - b] mod ({x^p - x})
ret = mod([a, zeros(1,numel(b)-numel(a))] - [b, zeros(1,numel(a)-numel(b))], p);
end

function ret = poly_mul_ring(a, b)
% polynomial multiplication in the ring Zp[x]
n1 = numel(a) - 1;
n2 = numel(b) - 1;
x = zeros(1, n1+n2+1);
for i = 1:n1+1
    for j = 1:n2+1
        x(i+j-1) = x(i+j-1) + a(i)*b(j);
    end
end
ret = x;
end

function [sig] = ntru_sign(message, sk)
% sk: private key
% f: public key 
% p: plaintext modulus
% q: ciphertext modulus

holds = sk{1};
g = sk{2};

% calculate polynomial h from message
h = mod(-1 * message * holds, sk{2});

% add random polynomial e to h
e = generate_poly(numel(h), sk{2});
h = mod(h + e, sk{2});

% calculate inverse of g
[~,~,inv_g] = poly_gcd_ring(g,sk{2});

% calculate signature
sig = mod(conv(h, inv_g), sk{2});

end

function [result] = ntru_verif(message, sig, pk)
% verifies NTRU signature--returns true if signature is valid, false otherwise.
% sk: private key
% f: public key 
% p: plaintext modulus
% q: ciphertext modulus

f_q = pk{1};
q = pk{2};

% calculate polynomial a from signature
a = mod(sig .* f_q, q);

% calculate inverse of f
[~,~,inv_f] = poly_gcd_ring(f_q,q);

% calculate polynomial b from message
b = mod(inv_f .* message, q);

% check if a = b
result = isequal(a,b);

end

% Test example
fprintf('Testing NTRU signature\n');

N = 10;
p = 3;
q = 7;

% Generate keys
[pk, sk] = ntru_keygen(N, p, q);

% Set message
message = [1, 0, 1, 1, 0, 1, 0, 0, 1, 1];

% Generate signature
sig = ntru_sign(message, sk);

% Verify signature
verif = ntru_verif(message, sig, pk);

if verif
    fprintf('Signature is valid!\n');
else
    error('Invalid signature');
end

请尝试使用上述代码进行测试，同时也欢迎您提出宝贵的意见和建议，帮助我们改进和优化代码。