previous/next safe prime

kryptools/__init__.py

@@ -9,7 +9,7 @@
 __email__ = "Gerald.Teschl@univie.ac.at"
 
 from .nt import egcd, cf, convergents, legendre_symbol, jacobi_symbol, sqrt_mod, euler_phi, carmichael_lambda, moebius_mu, is_carmichael_number, order, crt
-from .primes import sieve_eratosthenes, prime_pi, is_prime, next_prime, previous_prime, random_prime, random_strongprime, is_safeprime, random_safeprime, is_blumprime, random_blumprime, miller_rabin_test, lucas_test
+from .primes import sieve_eratosthenes, prime_pi, is_prime, next_prime, previous_prime, next_safeprime, previous_safeprime, random_prime, random_strongprime, is_safeprime, random_safeprime, is_blumprime, random_blumprime, miller_rabin_test, lucas_test
 from .intfuncs import iroot, ilog, perfect_square, perfect_power, prime_power
 from .factor import factorint, divisors
 from .dlp import dlog

kryptools/primes.py

@@ -5,6 +5,8 @@
     is_prime(n) test if n is probably prime
     next_prime(n) find the next prime larger or equal n
     previous_prime(n) find the previous prime smaller or equal n
+    next_safeprime(n) find the next safe prime larger or equal n
+    previous_safeprime(n) find the previous safe prime smaller or equal n
     random_bitlength(l) return a random number with bit lenght l
     random_prime(l) return a random prime with bit length l
     random_strongprime(l) return a random strong prime with factors of bit length l
@@ -188,7 +190,9 @@ def is_prime(n: int, trialdivision: bool = True) -> bool:
 
 def is_safeprime(p: int) -> bool:
     """Tests if a number is a safe prime."""
-    return is_prime(p) and is_prime((p - 1) // 2)
+    if p < 107:
+        return p in (5, 7, 11, 23, 47, 59, 83)
+    return p % 12 == 11 and is_prime(p) and is_prime((p - 1) // 2)
 
 def is_blumprime(p: int) -> bool:
     """Tests if a number is a Blum prime."""
@@ -218,6 +222,30 @@ def previous_prime(n: int) -> int:
         n -= 2
     return n
 
+def next_safeprime(n: int) -> int:
+    """Find the next safe prime larger or equal `n`."""
+    if n < 6:
+        return 5
+    if n < 8:
+        return 7
+    n = n - (n % 12) + 11  # make sure p % 12 = 11
+    while not is_safeprime(n):
+        n += 12
+    return n
+
+def previous_safeprime(n: int) -> int:
+    """Find the previous safe prime smaller or equal `n`."""
+    if n < 7:
+        return 5
+    if n < 11:
+        return 7
+    tmp = (n % 12) + 1
+    if tmp != 12:
+        n = n - tmp  # make sure p % 12 = 11
+    while not is_safeprime(n):
+        n -= 12
+    return n
+
 def random_bitlength(l: int) -> int:
     "Return a random number with bit lenght `l`."
     l = max(2, l)

tests/test_primes.py

@@ -1,7 +1,7 @@
 import pytest
 from math import isqrt
 from random import randint, seed
-from kryptools import sieve_eratosthenes, prime_pi, next_prime, previous_prime, is_prime, is_safeprime, is_blumprime
+from kryptools import sieve_eratosthenes, prime_pi, next_prime, previous_prime, next_safeprime, previous_safeprime, is_prime, is_safeprime, is_blumprime
 from kryptools import miller_rabin_test, lucas_test, random_prime, random_safeprime, random_blumprime, random_strongprime
 seed(0)
 
@@ -46,6 +46,8 @@ def sieve(max: int) -> list:
 def test_sieve_eratosthenes():
 	for i, p in enumerate(sieve_eratosthenes(OEIS_A000010[-1])):
 		assert p == OEIS_A000010[i]
+		assert previous_prime(p) == p
+		assert next_prime(p) == p
 		if i > 0:
 			assert previous_prime(p-1) == OEIS_A000010[i-1]
 		if i < len(OEIS_A000010) - 1:
@@ -74,8 +76,14 @@ def test_is_prime():
 def test_is_safeprime():
 	for n in range(OEIS_A005385[-1]+1):
 		assert is_safeprime(n) == (n in OEIS_A005385)
-	for n in OEIS_A005385:
-		assert is_safeprime(n) == True
+	for i, p in enumerate(OEIS_A005385):
+		assert is_safeprime(p) == True
+		assert previous_safeprime(p) == p
+		assert next_safeprime(p) == p
+		if i > 0:
+			assert previous_safeprime(p-1) == OEIS_A005385[i-1]
+		if i < len(OEIS_A005385) - 1:
+			assert next_safeprime(p+1) == OEIS_A005385[i+1]
 
 #https://oeis.org/A014233
 OEIS_A014233 = [ 2047, 1373653, 25326001, 3215031751, 2152302898747, 3474749660383,