Data update

Task/Tonelli-Shanks-algorithm/R/tonelli-shanks-algorithm.r

@@ -0,0 +1,98 @@
+library(gmp)  # For handling large integers and modular arithmetic
+
+# Legendre symbol: a^((p-1)/2) mod p
+legendre <- function(a, p) {
+  a <- as.bigz(a)
+  p <- as.bigz(p)
+  e <- div.bigz(sub.bigz(p, 1), 2)
+  result <- powm(a, e, p)
+  return(result)
+}
+
+# Tonelli-Shanks algorithm to find square root of n mod p
+tonelli <- function(n, p) {
+  n <- as.bigz(n)
+  p <- as.bigz(p)
+
+  # Check that n is a quadratic residue
+  leg <- legendre(n, p)
+  if (!identical(as.character(leg), "1")) {
+    stop("not a square (mod p)")
+  }
+
+  q <- sub.bigz(p, 1)
+  s <- 0
+  while (mod.bigz(q, 2) == 0) {
+    q <- div.bigz(q, 2)
+    s <- s + 1
+  }
+
+  if (s == 1) {
+    e <- div.bigz(add.bigz(p, 1), 4)
+    return(powm(n, e, p))
+  }
+
+  z <- as.bigz(2)
+  p_minus_1 <- sub.bigz(p, 1)
+  while (z < p) {
+    if (identical(as.character(legendre(z, p)), as.character(p_minus_1))) {
+      break
+    }
+    z <- add.bigz(z, 1)
+  }
+
+  c <- powm(z, q, p)
+  r <- powm(n, div.bigz(add.bigz(q, 1), 2), p)
+  t <- powm(n, q, p)
+  m <- s
+
+  one <- as.bigz(1)
+  two <- as.bigz(2)
+
+  while (!identical(mod.bigz(sub.bigz(t, one), p), as.bigz(0))) {
+    t2 <- mod.bigz(mul.bigz(t, t), p)
+    i <- 1
+    while (i < m) {
+      if (identical(mod.bigz(sub.bigz(t2, one), p), as.bigz(0))) {
+        break
+      }
+      t2 <- mod.bigz(mul.bigz(t2, t2), p)
+      i <- i + 1
+    }
+
+    b_exp <- pow.bigz(two, m - i - 1)
+    b <- powm(c, b_exp, p)
+    r <- mod.bigz(mul.bigz(r, b), p)
+    c <- mod.bigz(mul.bigz(b, b), p)
+    t <- mod.bigz(mul.bigz(t, c), p)
+    m <- i
+  }
+
+  return(r)
+}
+
+# Test cases
+ttest <- list(
+  c("10", "13"),
+  c("56", "101"),
+  c("1030", "10009"),
+  c("44402", "100049"),
+  c("665820697", "1000000009"),
+  c("881398088036", "1000000000039"),
+  c("41660815127637347468140745042827704103445750172002", "100000000000000000000000000000000000000000000000577")
+)
+
+for (case in ttest) {
+  n <- as.bigz(case[1])
+  p <- as.bigz(case[2])
+  r <- tonelli(n, p)
+  check <- mod.bigz(sub.bigz(mul.bigz(r, r), n), p)
+
+  # Verify the result
+  if (!identical(as.character(check), "0")) {
+    stop("Verification failed")
+  }
+
+  cat("n =", as.character(n), "p =", as.character(p), "\n")
+  cat("\troots :", as.character(r), as.character(sub.bigz(p, r)), "\n")
+}