帮我写一段用基础函数实现的NTRU数字签名功能的MATLAB代码

好的，我可以为您编写MATLAB代码来实现NTRU数字签名功能。以下是代码示例：

% NTRU数字签名功能的MATLAB代码
function [ signature ] = ntru_signature( message, private_key )

% 定义NTRU参数
N = 503; 
p = 3; 
q = 64; 
d = 45;
f = 3; 
g = 3; 

% 加载私钥
h = private_key.h;
f_inverse = private_key.f_inverse;

% 生成随机数P、R、E
P = randi([0,1],1,N);
R = randi([0,1],1,N);
E = randi([-1,0,1],1,N);

% 计算签名S
S = mod(p*f_inverse*(h.*R+mod(f.*P+g.*E,q)),q);

% 计算哈希值
hash = md5(message);

% 计算签名哈希
signature = mod(S+hash,q);

end

此代码实现了基础函数，使用NTRU参数和私钥对输入消息进行数字签名。注意：实际使用时，需要使用适当的参数和私钥生成函数来生成正确的参数和私钥。

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帮我写一段用基础函数实现的NTRU数字签名的MATLAB代码

好的，以下是基础函数实现的NTRU数字签名的MATLAB代码。

% Generate random keys for NTRU
N = 1024;
q = 2048; 
d = 3;
N0 = ceil(log2(q));
N1 = N - N0 - 1;
N2 = N - N1 - 1;
f = 1 + x - x^3 - x^4 - x^6 + x^8;
g = 3 + x^2 - x^3 - x^5 + x^6 + x^8;
h = 1 - x;

% Choose a random message for signing
m = randi([0 1], 1, N1);

% Hash message
hash = mod(sum(m.*2.^(0:N1-1)), q);

% Generate random ephemeral key for signing
e = randi([-1 1], 1, N2);

% Calculate signature
f_inv = PolyInvMod(f, q, d);
a = PolyMulMod(f_inv, g, q, d);
b = PolyMulMod(a, e, q, d);
c = PolyMulMod(a, f_inv, q, d);
d = mod(q - hash - PolyEvalMod(b, h, q) + 2 * floor(q/2), q);
z = PolyAddMod(d * c, e, q);

% Verify signature
w = PolySubMod(PolyEvalMod(b, f, q), PolyEvalMod(z, h, q), q);
if w == hash
    disp('VALID SIGNATURE');
else
    disp('INVALID SIGNATURE');
end

% Polynomial multiplication modulo q with polynomial modulus f
function z = PolyMulMod(x, y, q, d)
    z = mod(x * y, q);
    for i = 1:d-1
        z = PolyReduceMod(z, q);
    end
end

% Polynomial addition modulo q
function z = PolyAddMod(x, y, q)
    z = mod(x + y, q);
end

% Polynomial reduction modulo polynomial modulus f
function z = PolyReduceMod(x, q)
    f = 1 + x - x^3 - x^4 - x^6 + x^8;
    z = PolyMod(x, f, q);
end

% Polynomial division modulo q
function z = PolyDivMod(x, y, q)
    [q1, r] = deconv(x, y);
    z = mod(r, q);
end

% Polynomial modulo polynomial modulus f modulo q
function z = PolyMod(x, f, q)
    while length(x) >= length(f)
        if x(1) == 0
            x = x(2:end);
            continue;
        end
        shift = length(x) - length(f);
        factor = x(1) * f;
        factor = [zeros(1, shift), factor];
        x = mod(PolySubMod(x, factor, q), q);
    end
    z = x;
end

% Polynomial subtraction modulo q
function z = PolySubMod(x, y, q)
    z = mod(x - y, q);
end

% Inverse of polynomial x modulo q using Euclid's algorithm
function z = PolyInvMod(x, q, d)
    [g, u, v] = ExtendedEuclid(x, q);
    x_inv = mod(u * g^(d-1), q);
    z = PolyMod(x_inv, x, q);
end

% Extended Euclid's algorithm
function [g, u, v] = ExtendedEuclid(a, b)
    g0 = a; u0 = 1; v0 = 0;
    g1 = b; u1 = 0; v1 = 1;
    while g1 ~= 0
        q = floor(g0 / g1);
        temp = g0 - q * g1;
        u2 = u0 - q * u1;
        v2 = v0 - q * v1;
        g0 = g1;
        u0 = u1;
        v0 = v1;
        g1 = temp;
        u1 = u2;
        v1 = v2;
    end
    g = g0;
    u = u0;
    v = v0;
end

% Evaluate polynomial x modulo modulus m at point a using Horner's rule
function y = PolyEvalMod(x, a, m)
    y = 0;
    for i = length(x):-1:1
        y = mod(x(i) + a * y, m);
    end
end

帮我写一段用基础函数实现的比较低阶的NTRU数字签名的MATLAB代码

好的，以下是用基础函数实现的比较低阶的NTRU数字签名的MATLAB代码：

% Step 1: Generate public and private keys
N = 31; % Length of the polynomial
p = 3; % Modulus
q = 16; % Rescaling factor
f = genf(N,p,q); % Generate f
g = BinaryToString(fqm(f,q,N),p); % Compute g
F = fqm(f,p,N); % Generate F
h = genh(F,g,N); % Compute h
sk = ModReduce(randn(1,N),p); % Generate secret key
pk = [h,ModReduce(sk*g, p)]; % Generate public key

% Step 2: Sign a message
msg = 'Hello, world!'; % The message to be signed
K = BinaryToString(fqm(genf(N,2,1),2,N)); % Generate K, the random seed
r = ModReduce(fqm(GenRandomTernary(N),2,N),p); % Generate r
e = ModReduce(fqm(GenRandomTernary(N),2,N),p); % Generate e
A = ModReduce(fqm(qr(0,N-1),2,N),p); % Set A to the zero polynomial
B = ModReduce(sk*r + e + fqm(StringToBinary(msg),p,N)*A, p); % Compute B
InvF = ModInverse(F, p); % Compute the inverse of F modulo p
S = ModReduce(r + F*ModReduce(InvF*B - sk*e, p), p); % Compute the signature

% Step 3: Verify the signature
if (VerifySignature(S, StringToBinary(msg), pk, q, p, N) == 1)
disp('Signature verified successfully!')
else
disp('Signature verification failed!')
end

% ------------------ Helper functions ------------------

function f = genf(N, p, q)
f = zeros(1,N);
f(1) = 1;
f(N/2 + 1) = -1;
while true
f(2:N/2) = GenRandomTernary(N/2);
if (ModReduce(fqm(f,p,N),q,N) ~= 0)
break
end
end
end

function h = genh(F, g, N)
InvF = ModInverse(F, 3);
h = ModReduce(fqm(InvF*g,3,N),2,N);
end

function t = GenRandomTernary(N)
t = zeros(1,N);
nonzero_indices = randperm(N, ceil(N/3));
t(nonzero_indices) = randsample([-1 1], ceil(N/3), true);
end

function y = fqm(x, q, N)
y = ModReduce(q*ModReduce(x,2,N) - ModReduce(q-1,2,N),q,N);
end

function y = ModReduce(x, p)
y = mod(x,p);
y(y>p/2) = y(y>p/2) - p;
end

function y = ModInverse(x, p)
[gcd,inv1,inv2] = gcdex(x,p);
y = mod(inv1, p);
end

function y = BinaryToString(x, p)
y = char(mod(x,p) + 65);
end

function y = StringToBinary(x)
y = zeros(1,length(x));
for i = 1:length(x)
y(i) = x(i) - 65;
end
end

function y = VerifySignature(S, msg, pk, q, p, N)
InvF = ModInverse(fqm(pk(1:N),p,N),p);
inv_h_sk = ModReduce(fqm([pk(N+1:end),-pk(1:N)],p,N).*InvF,p,N);
A = ModReduce(fqm(-S,2,N).*InvF,p,N);
B = ModReduce(fqm(msg,2,N).*InvF,p,N);
lhs = ModReduce(fqm(pk(1:N),2,N).*A + fqm(pk(N+1:end),2,N).*B, p, N);
rhs = fqm(qr(0,N-1),q,N).*ModReduce(InvF*S,p,N);
y = isequal(lhs, rhs);
end