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— FCS (CRC): Polynomial —

Frame Check Sequence (FCS) in High-level Data Link Control (HDLC) frame used for the Point-to-Point Protocol (PPP)

There are a 16-bit and a 32-bit Frame Check Sequence methods used in the High-level Link Control frame as used in the Point-to-Point Protocol. The 16-bit FCS is the default, 32-bit FCS can be negotiated between the participating devices. Here, the 16-bit FCS is discussed only, the 32-bit one can be infered from this discussion, however.

Note that the FCS is sometimes called CRC (Cyclic Redundancy Check) for other applications than HDLC and PPP. They function in the same way, although the generator polynomial may be different.

What makes it somewhat hard to understand the FCS is the fact that it was originally designed for a hardware implementation. The bits drip along the serial telephone line. To speed up a purely software implementation uses bytes instead of bits, a precalculated table is used.

This generator polynomial applies to CCITT X.25 and UIT V.41 as well.

        x¹⁶ + x¹² + x⁵ + 1

Here is the table for this polynomial copied from RFC 1662, page 19:

	+ 0	+ 1	+ 2	+ 3	¦	+ 4	+ 5	+ 6	+ 7
00	0000	1189	2312	329b	¦	4624	57ad	6536	74bf	07
08	8c48	9dc1	af5a	bed3	¦	ca6c	dbe5	e97e	f8f7	0f
10	1081	0108	3393	221a	¦	56a5	472c	75b7	643e	17
18	9cc9	8d40	bfdb	ae52	¦	daed	cb64	f9ff	e876	1f
20	2102	308b	0210	1399	¦	6726	76af	4434	55bd	27
28	ad4a	bcc3	8e58	9fd1	¦	eb6e	fae7	c87c	d9f5	2f
30	3183	200a	1291	0318	¦	77a7	662e	54b5	453c	37
38	bdcb	ac42	9ed9	8f50	¦	fbef	ea66	d8fd	c974	3f
40	4204	538d	6116	709f	¦	0420	15a9	2732	36bb	47
48	ce4c	dfc5	ed5e	fcd7	¦	8868	99e1	ab7a	baf3	4f
50	5285	430c	7197	601e	¦	14a1	0528	37b3	263a	57
58	decd	cf44	fddf	ec56	¦	98e9	8960	bbfb	aa72	5f
60	6306	728f	4014	519d	¦	2522	34ab	0630	17b9	67
68	ef4e	fec7	cc5c	ddd5	¦	a96a	b8e3	8a78	9bf1	6f
70	7387	620e	5095	411c	¦	35a3	242a	16b1	0738	77
78	ffcf	ee46	dcdd	cd54	¦	b9eb	a862	9af9	8b70	7f
80	8408	9581	a71a	b693	¦	c22c	d3a5	e13e	f0b7	87
88	0840	19c9	2b52	3adb	¦	4e64	5fed	6d76	7cff	8f
90	9489	8500	b79b	a612	¦	d2ad	c324	f1bf	e036	97
98	18c1	0948	3bd3	2a5a	¦	5ee5	4f6c	7df7	6c7e	9f
a0	a50a	b483	8618	9791	¦	e32e	f2a7	c03c	d1b5	a7
a8	2942	38cb	0a50	1bd9	¦	6f66	7eef	4c74	5dfd	af
b0	b58b	a402	9699	8710	¦	f3af	e226	d0bd	c134	b7
b8	39c3	284a	1ad1	0b58	¦	7fe7	6e6e	5cf5	4d7c	bf
c0	c60c	d785	e51e	f497	¦	8028	91a1	a33a	b2b3	c7
c8	4a44	5bcd	6956	78df	¦	0c60	1de9	2f72	3efb	cf
d0	d68d	c704	f59f	e416	¦	90a9	8120	b3bb	a232	d7
d8	5ac5	4b4c	79d7	685e	¦	1ce1	0d68	3ff3	2e7a	df
e0	e70e	f687	c41c	d595	¦	a12a	b0a3	8238	93b1	e7
e8	6b46	7acf	4854	59dd	¦	2d62	3ceb	0e70	1ff9	ef
f0	f78f	e606	d49d	c514	¦	b1ab	a022	92b9	8330	f7
f8	7bc7	6a4e	58d5	495c	¦	3de3	2c6a	1ef1	0f78	ff
	+ 0	+ 1	+ 2	+ 3	¦	+ 4	+ 5	+ 6	+ 7

This table is available here as pure ascii-text files (2 KB) for

ready to copy into your source file.