diff --git a/backend/Cargo.lock b/backend/Cargo.lock
index 28444fe..2fdacb2 100644
--- a/backend/Cargo.lock
+++ b/backend/Cargo.lock
@@ -53,6 +53,7 @@ version = "0.1.0"
 dependencies = [
  "axum",
  "bcrypt",
+ "bytes",
  "chrono",
  "http-body-util",
  "jsonwebtoken",
@@ -138,6 +139,12 @@ dependencies = [
  "syn",
 ]
 
+[[package]]
+name = "base16ct"
+version = "0.2.0"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "4c7f02d4ea65f2c1853089ffd8d2787bdbc63de2f0d29dedbcf8ccdfa0ccd4cf"
+
 [[package]]
 name = "base64"
 version = "0.22.1"
@@ -309,6 +316,18 @@ version = "0.8.21"
 source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "d0a5c400df2834b80a4c3327b3aad3a4c4cd4de0629063962b03235697506a28"
 
+[[package]]
+name = "crypto-bigint"
+version = "0.5.5"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "0dc92fb57ca44df6db8059111ab3af99a63d5d0f8375d9972e319a379c6bab76"
+dependencies = [
+ "generic-array",
+ "rand_core 0.6.4",
+ "subtle",
+ "zeroize",
+]
+
 [[package]]
 name = "crypto-common"
 version = "0.1.7"
@@ -319,6 +338,33 @@ dependencies = [
  "typenum",
 ]
 
+[[package]]
+name = "curve25519-dalek"
+version = "4.1.3"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "97fb8b7c4503de7d6ae7b42ab72a5a59857b4c937ec27a3d4539dba95b5ab2be"
+dependencies = [
+ "cfg-if",
+ "cpufeatures",
+ "curve25519-dalek-derive",
+ "digest",
+ "fiat-crypto",
+ "rustc_version",
+ "subtle",
+ "zeroize",
+]
+
+[[package]]
+name = "curve25519-dalek-derive"
+version = "0.1.1"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "f46882e17999c6cc590af592290432be3bce0428cb0d5f8b6715e4dc7b383eb3"
+dependencies = [
+ "proc-macro2",
+ "quote",
+ "syn",
+]
+
 [[package]]
 name = "der"
 version = "0.7.10"
@@ -368,6 +414,44 @@ version = "0.15.7"
 source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "1aaf95b3e5c8f23aa320147307562d361db0ae0d51242340f558153b4eb2439b"
 
+[[package]]
+name = "ecdsa"
+version = "0.16.9"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "ee27f32b5c5292967d2d4a9d7f1e0b0aed2c15daded5a60300e4abb9d8020bca"
+dependencies = [
+ "der",
+ "digest",
+ "elliptic-curve",
+ "rfc6979",
+ "signature",
+ "spki",
+]
+
+[[package]]
+name = "ed25519"
+version = "2.2.3"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "115531babc129696a58c64a4fef0a8bf9e9698629fb97e9e40767d235cfbcd53"
+dependencies = [
+ "pkcs8",
+ "signature",
+]
+
+[[package]]
+name = "ed25519-dalek"
+version = "2.2.0"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "70e796c081cee67dc755e1a36a0a172b897fab85fc3f6bc48307991f64e4eca9"
+dependencies = [
+ "curve25519-dalek",
+ "ed25519",
+ "serde",
+ "sha2",
+ "subtle",
+ "zeroize",
+]
+
 [[package]]
 name = "either"
 version = "1.15.0"
@@ -377,6 +461,27 @@ dependencies = [
  "serde",
 ]
 
+[[package]]
+name = "elliptic-curve"
+version = "0.13.8"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "b5e6043086bf7973472e0c7dff2142ea0b680d30e18d9cc40f267efbf222bd47"
+dependencies = [
+ "base16ct",
+ "crypto-bigint",
+ "digest",
+ "ff",
+ "generic-array",
+ "group",
+ "hkdf",
+ "pem-rfc7468",
+ "pkcs8",
+ "rand_core 0.6.4",
+ "sec1",
+ "subtle",
+ "zeroize",
+]
+
 [[package]]
 name = "equivalent"
 version = "1.0.2"
@@ -415,6 +520,22 @@ dependencies = [
  "pin-project-lite",
 ]
 
+[[package]]
+name = "ff"
+version = "0.13.1"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "c0b50bfb653653f9ca9095b427bed08ab8d75a137839d9ad64eb11810d5b6393"
+dependencies = [
+ "rand_core 0.6.4",
+ "subtle",
+]
+
+[[package]]
+name = "fiat-crypto"
+version = "0.2.9"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "28dea519a9695b9977216879a3ebfddf92f1c08c05d984f8996aecd6ecdc811d"
+
 [[package]]
 name = "find-msvc-tools"
 version = "0.1.9"
@@ -526,6 +647,7 @@ checksum = "85649ca51fd72272d7821adaf274ad91c288277713d9c18820d8499a7ff69e9a"
 dependencies = [
  "typenum",
  "version_check",
+ "zeroize",
 ]
 
 [[package]]
@@ -564,6 +686,17 @@ dependencies = [
  "wasip3",
 ]
 
+[[package]]
+name = "group"
+version = "0.13.0"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "f0f9ef7462f7c099f518d754361858f86d8a07af53ba9af0fe635bbccb151a63"
+dependencies = [
+ "ff",
+ "rand_core 0.6.4",
+ "subtle",
+]
+
 [[package]]
 name = "hashbrown"
 version = "0.15.5"
@@ -892,11 +1025,18 @@ source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "0529410abe238729a60b108898784df8984c87f6054c9c4fcacc47e4803c1ce1"
 dependencies = [
  "base64",
+ "ed25519-dalek",
  "getrandom 0.2.17",
+ "hmac",
  "js-sys",
+ "p256",
+ "p384",
  "pem",
+ "rand 0.8.6",
+ "rsa",
  "serde",
  "serde_json",
+ "sha2",
  "signature",
  "simple_asn1",
 ]
@@ -1107,6 +1247,30 @@ version = "1.21.4"
 source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "9f7c3e4beb33f85d45ae3e3a1792185706c8e16d043238c593331cc7cd313b50"
 
+[[package]]
+name = "p256"
+version = "0.13.2"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "c9863ad85fa8f4460f9c48cb909d38a0d689dba1f6f6988a5e3e0d31071bcd4b"
+dependencies = [
+ "ecdsa",
+ "elliptic-curve",
+ "primeorder",
+ "sha2",
+]
+
+[[package]]
+name = "p384"
+version = "0.13.1"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "fe42f1670a52a47d448f14b6a5c61dd78fce51856e68edaa38f7ae3a46b8d6b6"
+dependencies = [
+ "ecdsa",
+ "elliptic-curve",
+ "primeorder",
+ "sha2",
+]
+
 [[package]]
 name = "parking"
 version = "2.2.1"
@@ -1234,6 +1398,15 @@ dependencies = [
  "syn",
 ]
 
+[[package]]
+name = "primeorder"
+version = "0.13.6"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "353e1ca18966c16d9deb1c69278edbc5f194139612772bd9537af60ac231e1e6"
+dependencies = [
+ "elliptic-curve",
+]
+
 [[package]]
 name = "proc-macro2"
 version = "1.0.106"
@@ -1358,6 +1531,16 @@ version = "0.8.10"
 source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "dc897dd8d9e8bd1ed8cdad82b5966c3e0ecae09fb1907d58efaa013543185d0a"
 
+[[package]]
+name = "rfc6979"
+version = "0.4.0"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "f8dd2a808d456c4a54e300a23e9f5a67e122c3024119acbfd73e3bf664491cb2"
+dependencies = [
+ "hmac",
+ "subtle",
+]
+
 [[package]]
 name = "rsa"
 version = "0.9.10"
@@ -1378,6 +1561,15 @@ dependencies = [
  "zeroize",
 ]
 
+[[package]]
+name = "rustc_version"
+version = "0.4.1"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "cfcb3a22ef46e85b45de6ee7e79d063319ebb6594faafcf1c225ea92ab6e9b92"
+dependencies = [
+ "semver",
+]
+
 [[package]]
 name = "rustversion"
 version = "1.0.22"
@@ -1396,6 +1588,20 @@ version = "1.2.0"
 source = "registry+https://github.com/rust-lang/crates.io-index"
 checksum = "94143f37725109f92c262ed2cf5e59bce7498c01bcc1502d7b9afe439a4e9f49"
 
+[[package]]
+name = "sec1"
+version = "0.7.3"
+source = "registry+https://github.com/rust-lang/crates.io-index"
+checksum = "d3e97a565f76233a6003f9f5c54be1d9c5bdfa3eccfb189469f11ec4901c47dc"
+dependencies = [
+ "base16ct",
+ "der",
+ "generic-array",
+ "pkcs8",
+ "subtle",
+ "zeroize",
+]
+
 [[package]]
 name = "semver"
 version = "1.0.28"
diff --git a/backend/Cargo.toml b/backend/Cargo.toml
index ca4aeb4..14f862c 100644
--- a/backend/Cargo.toml
+++ b/backend/Cargo.toml
@@ -10,7 +10,7 @@ tokio = { version = "1", features = ["full"] }
 sqlx = { version = "0.8", features = ["sqlite", "runtime-tokio", "macros", "migrate"] }
 serde = { version = "1", features = ["derive"] }
 serde_json = "1"
-jsonwebtoken = "10"
+jsonwebtoken = { version = "10", features = ["rust_crypto"] }
 bcrypt = "0.19"
 tower-http = { version = "0.6", features = ["fs", "cors"] }
 chrono = { version = "0.4", features = ["serde"] }
@@ -22,3 +22,4 @@ tracing-subscriber = { version = "0.3", features = ["env-filter"] }
 [dev-dependencies]
 tower = { version = "0.5", features = ["util"] }
 http-body-util = "0.1"
+bytes = "1"
diff --git a/backend/src/auth.rs b/backend/src/auth.rs
index 1df4b0b..db87274 100644
--- a/backend/src/auth.rs
+++ b/backend/src/auth.rs
@@ -1 +1,56 @@
-// auth — populated in Task 4
+use axum::{extract::FromRequestParts, http::request::Parts};
+use jsonwebtoken::{decode, encode, DecodingKey, EncodingKey, Header, Validation};
+use serde::{Deserialize, Serialize};
+use crate::error::AppError;
+
+#[derive(Debug, Serialize, Deserialize, Clone)]
+pub struct TutorClaims { pub sub: i64, pub email: String, pub exp: usize }
+
+fn secret() -> String {
+    std::env::var("JWT_SECRET").expect("JWT_SECRET not set")
+}
+
+pub fn encode_jwt(id: i64, email: &str) -> Result<String, AppError> {
+    let exp = chrono::Utc::now().timestamp() as usize + 86400 * 7;
+    let claims = TutorClaims { sub: id, email: email.into(), exp };
+    encode(&Header::default(), &claims, &EncodingKey::from_secret(secret().as_bytes()))
+        .map_err(|_| AppError::Unauthorized)
+}
+
+pub fn decode_jwt(token: &str) -> Result<TutorClaims, AppError> {
+    decode::<TutorClaims>(token, &DecodingKey::from_secret(secret().as_bytes()),
+        &Validation::default())
+        .map(|d| d.claims)
+        .map_err(|_| AppError::Unauthorized)
+}
+
+// Axum extractor: pulls JWT from Authorization: Bearer header
+impl<S: Send + Sync> FromRequestParts<S> for TutorClaims {
+    type Rejection = AppError;
+    async fn from_request_parts(parts: &mut Parts, _: &S) -> Result<Self, Self::Rejection> {
+        let header = parts.headers.get("authorization")
+            .and_then(|v| v.to_str().ok())
+            .and_then(|v| v.strip_prefix("Bearer "))
+            .ok_or(AppError::Unauthorized)?;
+        decode_jwt(header)
+    }
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+
+    #[test]
+    fn roundtrip_jwt() {
+        unsafe { std::env::set_var("JWT_SECRET", "testsecret"); }
+        let token = encode_jwt(1, "test@example.com").unwrap();
+        let claims = decode_jwt(&token).unwrap();
+        assert_eq!(claims.sub, 1);
+    }
+
+    #[test]
+    fn invalid_jwt_rejected() {
+        unsafe { std::env::set_var("JWT_SECRET", "testsecret"); }
+        assert!(decode_jwt("not.a.token").is_err());
+    }
+}
diff --git a/backend/src/main.rs b/backend/src/main.rs
index 2209dfd..f0545f7 100644
--- a/backend/src/main.rs
+++ b/backend/src/main.rs
@@ -4,6 +4,9 @@ mod models;
 mod auth;
 mod routes;
 
+#[cfg(test)]
+mod test_helpers;
+
 use axum::Router;
 use tracing_subscriber::EnvFilter;
 
diff --git a/backend/src/routes/auth_routes.rs b/backend/src/routes/auth_routes.rs
new file mode 100644
index 0000000..767fe48
--- /dev/null
+++ b/backend/src/routes/auth_routes.rs
@@ -0,0 +1,62 @@
+use axum::{extract::State, routing::post, Json, Router};
+use serde::Deserialize;
+use sqlx::SqlitePool;
+use serde_json::{json, Value};
+use crate::{auth, error::AppError};
+
+#[derive(Deserialize)]
+struct LoginRequest { email: String, password: String }
+
+async fn login(
+    State(pool): State<SqlitePool>,
+    Json(req): Json<LoginRequest>,
+) -> Result<Json<Value>, AppError> {
+    let tutor: Option<(i64, String, String)> = sqlx::query_as(
+        "SELECT id, email, password_hash FROM tutors WHERE email = ?"
+    ).bind(&req.email).fetch_optional(&pool).await?;
+
+    let (id, email, hash) = tutor.ok_or(AppError::Unauthorized)?;
+    if !bcrypt::verify(&req.password, &hash).unwrap_or(false) {
+        return Err(AppError::Unauthorized);
+    }
+    let token = auth::encode_jwt(id, &email)?;
+    Ok(Json(json!({"token": token})))
+}
+
+pub fn router() -> Router<SqlitePool> {
+    Router::new().route("/api/auth/login", post(login))
+}
+
+#[cfg(test)]
+mod tests {
+    use super::*;
+    use crate::test_helpers::{build_test_app, post_json};
+    use serde_json::json;
+
+    #[sqlx::test(migrations = "./migrations")]
+    async fn login_returns_token(pool: SqlitePool) {
+        let hash = bcrypt::hash("secret", bcrypt::DEFAULT_COST).unwrap();
+        sqlx::query("INSERT INTO tutors (name,email,password_hash) VALUES (?,?,?)")
+            .bind("Test").bind("t@test.com").bind(&hash)
+            .execute(&pool).await.unwrap();
+
+        let app = crate::routes::build(pool);
+        let (status, body) = post_json(app, "/api/auth/login", "",
+            json!({"email":"t@test.com","password":"secret"})).await;
+        assert_eq!(status, 200);
+        assert!(serde_json::from_slice::<Value>(&body).unwrap()["token"].is_string());
+    }
+
+    #[sqlx::test(migrations = "./migrations")]
+    async fn login_wrong_password(pool: SqlitePool) {
+        let hash = bcrypt::hash("correct", bcrypt::DEFAULT_COST).unwrap();
+        sqlx::query("INSERT INTO tutors (name,email,password_hash) VALUES (?,?,?)")
+            .bind("Test").bind("t@test.com").bind(&hash)
+            .execute(&pool).await.unwrap();
+
+        let app = crate::routes::build(pool);
+        let (status, _) = post_json(app, "/api/auth/login", "",
+            json!({"email":"t@test.com","password":"wrong"})).await;
+        assert_eq!(status, 401);
+    }
+}
diff --git a/backend/src/routes/mod.rs b/backend/src/routes/mod.rs
index 3197a96..10e0bf1 100644
--- a/backend/src/routes/mod.rs
+++ b/backend/src/routes/mod.rs
@@ -1,6 +1,10 @@
 use axum::Router;
 use sqlx::SqlitePool;
 
-pub fn build(_pool: SqlitePool) -> Router {
+mod auth_routes;
+
+pub fn build(pool: SqlitePool) -> Router {
     Router::new()
+        .merge(auth_routes::router())
+        .with_state(pool)
 }
diff --git a/backend/src/test_helpers.rs b/backend/src/test_helpers.rs
new file mode 100644
index 0000000..7ff5136
--- /dev/null
+++ b/backend/src/test_helpers.rs
@@ -0,0 +1,55 @@
+// cfg(test) only — this whole module is test-only
+use sqlx::SqlitePool;
+use axum::Router;
+use tower::ServiceExt;
+use axum::http::{Request, StatusCode};
+use http_body_util::BodyExt;
+
+/// Insert a test tutor (if not exists), return a valid JWT for that tutor.
+pub async fn make_token(pool: &SqlitePool, email: &str) -> String {
+    let hash = bcrypt::hash("testpass", bcrypt::DEFAULT_COST).unwrap();
+    sqlx::query("INSERT OR IGNORE INTO tutors (name,email,password_hash) VALUES (?,?,?)")
+        .bind("Test Tutor").bind(email).bind(&hash)
+        .execute(pool).await.unwrap();
+    let id: (i64,) = sqlx::query_as("SELECT id FROM tutors WHERE email = ?")
+        .bind(email).fetch_one(pool).await.unwrap();
+    unsafe { std::env::set_var("JWT_SECRET", "testsecret"); }
+    crate::auth::encode_jwt(id.0, email).unwrap()
+}
+
+/// Build the full Axum app wired with the given pool, plus a Bearer auth header value.
+pub async fn build_test_app(pool: SqlitePool) -> (Router, String) {
+    unsafe { std::env::set_var("JWT_SECRET", "testsecret"); }
+    let token = make_token(&pool, "tutor@test.com").await;
+    let app = crate::routes::build(pool);
+    (app, format!("Bearer {token}"))
+}
+
+/// POST JSON body to the app (one-shot), returns (StatusCode, response body bytes).
+pub async fn post_json(app: Router, path: &str, auth: &str, body: serde_json::Value)
+    -> (StatusCode, bytes::Bytes)
+{
+    let req = Request::builder()
+        .method("POST").uri(path)
+        .header("Content-Type", "application/json")
+        .header("Authorization", auth)
+        .body(axum::body::Body::from(body.to_string()))
+        .unwrap();
+    let res = app.oneshot(req).await.unwrap();
+    let status = res.status();
+    let body = res.into_body().collect().await.unwrap().to_bytes();
+    (status, body)
+}
+
+/// GET from the app (one-shot), returns (StatusCode, response body bytes).
+pub async fn get(app: Router, path: &str, auth: &str) -> (StatusCode, bytes::Bytes) {
+    let req = Request::builder()
+        .method("GET").uri(path)
+        .header("Authorization", auth)
+        .body(axum::body::Body::empty())
+        .unwrap();
+    let res = app.oneshot(req).await.unwrap();
+    let status = res.status();
+    let body = res.into_body().collect().await.unwrap().to_bytes();
+    (status, body)
+}